MACHINE  DESIGN 


BY 
CHARLES  H.  BENJAMIN, 

DEAN  OP  THE  SCHOOLS  OF  ENGINEERING,  PUBDUE  UNIVERSITY 

AND 

JAMES  D.  HOFFMAN 

PKOFESSOB  OF  MECHANICAL  ENGINEEBING  AND  PBACTICAL 
MECHANICS,  UNIVEBSITY  OF  NEBBASKA 


NEW  YORK 
HENRY  HOLT  AND  COMPANY 

1913 


-^ 


COPYBIGHT,    1909 

BY 
JAMES  D.  HOFFMAN 


COPYRIGHT,  1906,  1913 

BY 
HENRY  HOLT  AND  COMPANY 


THE. MAPLE. PRESS. YORK. PA 


PREFACE 

The  present  book  represents  the  consolidation  of  two  texts 
on  this  subject,  Benjamin's  Machine  Design  and  Hoffman's 
Elementary  Machine  Design. 

As  now  arranged,  the  book  serves  two  purposes:  That  of  a 
text  for  the  classroom,  embodying  the  theory  and  practice  of 
design,  and  that  of  a  reference  book  for  the  drafting  room, 
illustrating  the  design  of  complete  machines. 

The  authors  recognize  the  fact  that  there  are  two  methods  of 
teaching  this  subject,  one  by  details  separately  treated  as  ele- 
ments, one  by  a  consideration  of  the  complete  machine,  i.e.,  one 
method  is  synthetic  and  one  analytic.  It  is  believed  that  this 
book  will  afford  a  means  of  using  either  method  or  both  combined. 

Some  important  additions  to  the  text  are  worthy  of  mention. 
Chapter  II,  on  Materials,  has  been  rewritten.  Much  additional 
matter  on  the  subjec^  of  cast-iron  frames  has  been  introduced, 
involving  the  results  of  numerous  experiments.  The  theoretical 
and  experimental  strength  of  steel  tubes  under  collapsing  pres- 
sures is  quite  fully  discussed  and  additional  data  are  given  on 
the  failure  of  pipe  fittings. 

Other  subjects  which  receive  in  this  volume  fuller  treatment 
than  heretofore  are  Flat  plates,  Crane  hooks,  Leaf  springs,  Bear- 
ings, both  plain  and  rolling,  Clutches,  Gear  teeth  and  Belting. 


in 


TABLE  OF  CONTENTS 

CHAPTER  PAGE 

INTRODUCTION. — UNITS  AND  FORMULAS 1 

1.  Units.     2.  Abbreviations.     3.  Notation.      4.  Formulas.     5. 

Profiles  of  uniform  strength. 

I.  MATERIALS 9 

6.  Primary  classification.  7.  Iron.  8.  Steel.  9.  Steel  alloys. 
10.  Copper  alloys.  11.  Strength  and  Elasticity. 

II.  FRAME  DESIGN 21 

12.  General  principles  of  design.  13.  Machine  supports.  14. 
Machine  frames.  15.  Tests  on  simple  beams.  16.  Shapes  of 
frames.  17.  Stresses  in  frames.  18.  Professor  Jenkin's  ex- 
periments. 19.  Purdue  tests.  20.  Principles  of  design. 

III.  CYLINDERS  AND  PIPES 50 

21.  Thin  shells.     22.  Thick  shells.     23.  Steel  and  wrought  iron 
pipe.     24.  Strength  of  boiler  tubes.     25.   Theory.     26.    Tube 
joints.     27.   Tubes   under  concentrated  loads.     28.   Pipe    fit- 
tings.    29.  Flanged  pittings.     30.  Steam  cylinders.     31.  Thick- 
ness of  flat  plates.     32.  Steel  plates. 

IV.  FASTENINGS 91 

33.  Bolts   and   nuts.     34.  Crane   hooks.     35.  Riveted   joints. 

36.  Lap  joints.  37.  Butt  joint  with  two  straps.  38.  Effi- 
ciency of  joints.  39.  Butt  joints  with  unequal  straps.  40. 
Practical  rules.  41.  Riveted  joints  for  narrow  plates.  42. 
Joint  pins.  43.  Cotters. 

V.  SPRINGS 107 

44.  Helical  springs.  45.  Square  wire.  46.  Experiments.  47. 
Springs  in  torsion.  48.  Flat  springs.  49.  Elliptic  and  semi- 
elliptic  springs. 

VI.  SLIDING  BEARINGS 120 

50.  Slides  in  general.  51.  Angular  slides.  52.  Gibbed  slides. 
53.  Flat  slides.  54.  Circular  guides.  55.  Stuffing  boxes. 

VII.  JOURNALS,  PIVOTS  AND  BEARINGS 128 

56.  Journals.  57.  Adjustment.  58.  Lubrication.  59.  Fric- 
tion of  journals.  60.  Limits  of  pressure.  61.  Heating  of  jour- 
nals. 62.  Experiments.  63.  Strength  and  stiffness  of  jour- 
nals. 64.  Caps  and  bolts.  65.  Step  bearings.  66.  Friction  of 
pivots.  67.  Flat  collar.  68.  Conical  pivot.  69.  Schiele's 
pivot.  70.  Multiple  bearing. 


vi  TABLE  OF  CONTENTS 

CHAPTER  PAGE 

VIII.   BALL  AND  ROLLER  BEARINGS 153 

71.  General  principles.  72.  Journal  bearings.  73.  Step  bear- 
ings. 74.  Materials  and  wear.  75.  Design  of  bearings.  76. 
Endurance  of  ball  bearings.  77.  Roller  bearings.  78.  Grant 
roller  bearings.  79.  Hyatt  rollers.  80.  Roller  step  bearings. 

81.  Design  of  roller  bearings. 

IX.  SHAFTING,  COUPLINGS  AND  HANGERS 167 

82.  Strength  of  shafting.     83.   Combined  tension  and  bending. 
84.  Couplings.     85.  Clutches.     86.  Automobile    clutches.     87. 
Coupling   bolts.     88.   Shafting   keys.     89.   Strength    of   keyed 
shafts.     90.  Hangers  and  boxes. 

X.  GEARS,  PULLEYS  AND  CRANKS 186 

91.  Gear  teeth.  92.  Strength  of  gear  teeth.  93.  Lewis'  for- 
mula. 94.  Experimental  data.  95.  Modern  practice.  96. 
Teeth  of  bevel  gears.  97.  Rim  and  arms.  98.  Sprocket  wheels 
and  chains.  99.  Silent  chains.  100.  Cranks  and  levers. 

XL  FLY-WHEELS 204 

101.  In  general.  102.  Safe  speed  for  wheels.  103.  Experi- 
ments on  fly-wheels.  104.  Wooden  pulleys.  105.  Rims  of  cast- 
iron  gears.  106.  Rotating  discs.  107.  Plain  discs.  108.  Con- 
ical discs.  109.  Discs  with  logarithmic  profile.  110.  Bursting 
speeds.  111.  Tests  of  discs. 

XII.  TRANSMISSION  BY  BELTS  AND  ROPES 221 

112.  Friction  of  belting.  113.  Slip  of  belt.  114.  Coefficient 
of  friction.  115.  Strength  of  belting.  116.  Taylor's  experi- 
ments. 117.  Rules  for  width  of  belts.  118.  Speed  of  belting. 
119.  Manila  rope  transmission.  120.  Strength  of  Manila  ropes. 
121.  Cotton  rope  transmission.  122.  Wire  rope  transmission. 

XIII.  DESIGN  OF  TOGGLE-JOINT  PRESS 235 

123.    Introductory.     124.    Drawings.    125.    Calculations.     126. 
Analysis    of    forces.     127.  Design    of    lever.     128.    Shapes    of 
levers.     129.  Hole    in    lever.     130.  Fastening     for     standard. 
131.  Design  of  standard.     132.  Toggle  joint.     133.  Shapes   of 
toggle  members.     134.  Die  heads.     135.  Frame  or  bed.     136. 
Stability  of    frame.     137.  Toggle  press.     138.   Vertical   hand- 
power  press.    139.  Foot-power  press.    140.  Hand-power  punch. 

141.  Punch  and  shear. 

XIV.  DESIGN  OF  BELT-DRIVEN  PUNCH  OR  SHEAR 265 

142.  General  statement.     143.  Requirements  of  design.     144. 
Design  of  frame.     145.  Outline  of  frame.     146.  Shearing  force. 
147.  Depth    of   cut.     148.  Sizes    of   pulleys.     149.  Weight    of 
fly-wheel.     150.  Driving  shaft.     151.  Gears.     152.  Main  shaft. 
153.  Sliding   head.     154.  Clutches   and    transmission   devices. 


TABLE  OF  CONTENTS  vii 

• 

CHAPTER  PAGE 

155.  Punch,  die  and  holders.  156.  The.  bevel  shear.  157. 
Horizontal  power  punch.  158.  The  bull-dozer.  159.  Power 
press.  160.  Rotary  shear.  161.  Sheet  metal  flanger.  162. 
Flanging  machine. 

XV.  DESIGN  OF  Am  HOIST  AND  RIVETER 299 

163.  Air  hoist.  164,  165.  Portable  riveter.  166.  Alligator 
riveter.  167.  Mud-ring  riveter.  168.  Lever  riveter.  169. 
Hydraulic  riveter.  170.  Triple  pressure  hydraulic  riveter. 

XVI.  STUDIES  IN  THE  KINEMATICS  OF  MACHINES 309 

171.  Gear  tooth  outlines.  172.  Planer  cam.  173.  Sewing 
machine  cam.  174.  Sewing  machine  bobbin  winder.  175. 
Constant  diameter  cam.  176,  177,  178,  179.  Quick  return 
motions.  180.  Cam  and  oscillating  arm.  181.  Two-motion 
cam.  182.  Conical  cam.  183,  184.  Motion  problems.  185. 
Forming  machine  for  wire  clips.  186-202.  Motion  problems. 
203.  Mechanism  of  inertia  governor.  204.  Mechanism  of 
centrifugal  governors.  205.  Straight  line  governor.  206. 
Stephenson  link  motion.  207.  Walschaert  valve  gear. 
INDEX .  336 


MACHINE  DESIGN 


TABLES 

TABLE  PAGE 

I.  VALUES  OP  Q  IN   COLUMN  FORMULA 4 

I  a.  VALUES  OF  S  AND  K  IN  COLUMN  FORMULA    ......  5 

II.  CONSTANTS  OF  CROSS-SECTION 6 

III.  FORMULAS  FOR  LOADED  BEAMS 7 

IV.  CLASSIFICATION  OF  METALS 9 

V.  COMPOSITION  OF  BRONZES 16 

VI.  STRENGTH  OF  WROUGHT  METALS .    .  18 

VII.  STRENGTH  OF  CAST  METALS 19 

VIII.  STRENGTH  OF  CAST  IRON  BEAMS 30 

IX.  STRENGTH  OF  CAST  IRON  BEAMS 32 

X.  STRENGTH  OF  CAST  IRON  BEAMS 33 

XI.  STRENGTH  OF  CAST  IRON  BEAMS 35 

XII.  STRESSES  IN  MACHINE  FRAMES 38 

XIII.  STRENGTH  OF  RIVETER  FRAMES 41 

XIV.  STRENGTH  OF  RIVETER  FRAMES 44 

XV.  SIZES  OF  IRON  AND  STEEL  PIPE 56 

XVI.  SIZES  OF  EXTRA  STRONG  PIPE 58 

XVII.  SIZES  OF  DOUBLE  EXTRA  STRONG  PIPE  ........  59 

XVIII.  SIZES  OF  IRON  AND  STEEL  BOILER  TUBES 60 

XIX.  COLLAPSING  PRESSURE  OF  TUBES 64 

XX.  STIFFNESS  OF  STEEL  HOOPS 70 

XXI.  STRENGTH  OF  STANDARD  SCREWED  PIPE  FITTINGS    ...  72 

XXII.  STRENGTH  OF  FLANGED  FITTINGS      74 

XXIII.  BURSTING  STRENGTH  OF  CAST  IRON  CYLINDERS    ....  80 

XXIV.  STRENGTH  OF  REINFORCED  CYLINDERS 82 

XXV.  STRENGTH  OF  CAST  IRON  PLATES 86 

XXVI.  STRENGTH  OF  CAST  IRON  PLATES 87 

XXVII.  STRESSES  IN  FLAT  PLATES      89 

XXVIII.  STRENGTH  OF  IRON  OR  STEEL  BOLTS 91 

XXIX.  DIMENSIONS  OF  MACHINE  SCREWS 94 

XXX.  ELASTIC  LIMIT  OF  CRANE  HOOKS 94 

XXXI.  DIMENSIONS  OF  RIVETED  LAP  JOINTS 101 

XXXII.  DIMENSIONS  OF  RIVETED  BUTT  JOINTS       101 

XXXIII.  STRENGTH  AND  STIFFNESS  OF  HELICAL  SPRINGS  .    .    .    .110 

XXXIV.  FRICTION  OF  PISTON  ROD  PACKINGS 126 

XXXV.  FRICTION  OF  PISTON  ROD  PACKINGS 126 

XXXVI.  FRICTION  OF  PISTON  ROD  PACKINGS 127 

XXXVII.  FRICTION  OF    JOURNAL  BEARINGS .    .   139 

XXXVIII.  TESTS  OF  LARGE  JOURNALS 140 

ix 


PLATES 

TABLE  PAGE 

XXXIX.  MARINE  THRUST  BEARINGS 151 

XL.  FRICTION  OF  ROLLER  AND  PLAIN  BEARINGS 161 

XLI.  COEFFICIENTS  OF  FRICTION 161 

XLII.  COEFFICIENTS  OF  FRICTION 162 

XLIII.  SAFE  LOADS  FOR  ROLLER  BEARINGS 164 

XLIV.  SAFE  LOADS  FOR  ROLLER  STEP  BEARINGS 165 

XLV.  VALUES  OF  K  FOR  ROLLER  THRUST  BEARINGS      .    .    .    .166 

XL VI.  DIAMETERS  OF  SHAFTING 168 

XL VII.  POWER  OF  CLUTCHES 177 

XLVIII.  EFFICIENCY  OF  KEYED  SHAFTS      181 

XLIX.  PROPORTIONS  OF  GEAR  TEETH 187 

L.  SIZES  OF  TEST  FLY-WHEELS 208 

LI.  SIZES  OF  TEST  FLY-WHEELS 209 

LII.  FLANGES  AND  BOLTS  OF  TEST  FLY-WHEELS 209 

LIII.  FAILURE  OF  FLANGED  JOINTS 210 

LIV.  SIZES  OF  LINKED  JOINTS 210 

LV.  FAILURE  OF  LINKED  JOINTS       210 

LVI.  BURSTING  SPEEDS  OF  ROTATING  Discs 218 

LVII.  BURSTING  SPEEDS  OF  ROTATING  Discs 219 

LVIII.  HORSE-POWER  OF  MANILA  ROPE       231 

LIX.  HORSE-POWER  OF  COTTON  ROPE 232 

LX.  HCRSE-POWER  OF  WIRE  ROPE  .   233 


PLATES 

PLATES  PAGE 

C  1.  TOGGLE  JOINT  PRESS  ASSEMBLY Facing  236 

C  2.  TOGGLE  JOINT  PRESS  DETAILS  . Facing  238 

C  3.  TOGGLE  JOINT  PRESS  DETAILS Facing  240 

C  4.  SINGLE  POWER  PUNCH  ASSEMBLY 287 

C  5.  SINGLE  POWER  PUNCH  DETAILS 288 

C  6.  SINGLE  POWER  PUNCH  DETAILS 289 

C  7.  SINGLE  POWER  PUNCH  DETAILS Facing  290 

C  8.  SINGLE  POWER  PUNCH  DETAILS    .  .   290 


MACHINE  DESIGN 


INTRODUCTION 
UNITS  AND  FORMULAS 

1.  Units. — In  this  book  the  following  units  will  be  used  unless 
otherwise  stated. 

Dimensions  in  inches. 

Forces  in  pounds. 

Stresses  in  pounds  per  square  inch. 

Velocities  in  feet  per  second. 

Work  and  energy  in  foot  pounds. 

Moments  in  pounds  inches. 

Speeds  of  lotation  in  revolutions  per  minute. 

The  word  stress  will  be  used  to  denote  the  resistance  of  material 
to  distortion  per  unit  of  sectional  area.  The  word  deformation 
will  be  used  to  denote  the  distortion  of  a  piece  per  unit  of  length. 
The  word  set  will  be  used  to  denote  total  permanent  distortion. 

In  making  calculations  the  use  of  the  slide-rule  and  of  four- 
place  logarithms  is  recommended;  accuracy  is  expected  only  to 
three  significant  figures. 

2.  Abbreviations. — The  following  abbreviations  are  among 
those  recommended  by  a  committee  of  the  American  Society  of 
Mechanical  Engineers  in  December,  1904,  and  will  be  used 
throughout  the  book.1 

NAME  ABBREVIATION 

Inches in. 

Feet ft. 

Yards yd. 

1  Tr.  A.  S.  M.  E.,  Vol.  XXVI,  p.  60. 

1 


2'  MACHINE  DESIGN 

NAME  ABBREVIATION 

Miles spell  out. 

Pounds lb. 

Tons spell  out. 

Gallons       gal. 

Seconds      sec. 

Minutes      min. 

Hours hr. 

Linear .  lin. 

Square sq. 

Cubic cu. 

Per spell  out. 

Fahrenheit fahr. 

Percentage %  or  per  cent. 

Revolutions  per  minute r.p.m. 

Brake  horse  power b.h.p. 

Electric  horse  power e.h.p. 

Indicated  horse  power i.h.p. 

British  thermal  units B.t.u. 

Diameter Diam. 

3.  Notation. 

Arc  of  contact  =  6  radians. 

Area  of  section  =A  sq.  in. 

Breadth  of  section  =b  in. 

Coefficient  of  friction  =/ 

Deflection  of  beam  —  A  in. 

Degrees  —  deg. 

Depth  of  section  =h  in. 

Diameter  of  circular  section  =d  in. 

Distance  of  neutral  axis  from  outer  fiber  =y  in. 

Elasticity,  modulus  of, 

in  tension  and  compression  —  E 

in  shearing  and  torsion  =G 

Heaviness,  weight  per  cu.  ft.  =w 

Length  of  any  member  =1  in. 

Load  or  dead  weight  =  W  lb. 

Moment,  in  bending  =  M  Ib.-in. 

in  twisting  =  T  Ib.-in. 


FORMULAS  3 

t 

Moment  of  inertia 

rectangular  =7 

polar  =«/ 

Pitch  of  teeth,  rivets,  etc.  =p  in. 

Radius  of  gyration  =r  in. 

Section  modulus,  bending  =-=Z 

twisting  =-=Zp 

y 

Stress  per  unit  of  area  =  S 

Velocity  in  feet  per  second  =v  ft.  per  sec. 


4.  Formulas. 


Simple  Stress 


W 

Tension,  compression  or  shear,  S=—r  (1) 

A. 

Bending  under  Transverse  Load 

SI 

General  equation,  M  =  —  (2) 

y 

Rectangular  section,  M  =  —  -  —  (3) 

Rectangular  section,  bh2  =  —  ~—  (4) 

Circular  section,  M  =  —^-^>  (5) 

Circular  section, 

Torsion  or  Twisting 

Q    T 

General  equation,  T=  —  (7) 

\s 


Circular  section,  7T  =  —  —  -.  (8) 

o.l 

3  /^   ^  rp 

Circular  section,  d  =  \—  ^  —  (9) 

o 

Hollow  circular  section,  T  =  ~  -  -         1  .          (10) 

o.l        d 

Other  values  of  -  and  -  may  be  taken  from  Table  II. 

y       y 


MACHINE  DESIGN 


Combined  Bending  and  Twisting 
Calculate  shaft  for  a  bending  moment, 


Column  subject  to  Bending 


W        S 
Use  Rankine's  formula,     -  = 


(11) 


(12) 


The  values  of  r2  may  be  taken  from  Table  II.  The  subjoined 
table  gives  the  average  values  of  g,  while  S  is  the  compressive 
strength  of  the  material. 

TABLE  I 

VALUES  OP  q  IN  FORMULA  12 


Material 

Both  ends 
fixed 

Fixed  and 
round 

Both  ends 
round 

Fixed  and 
free 

TimV»pr 

1 

1.78 

4 

16 

3000 
1 

3000 
1.78 

3000 
4 

3000 
16 

Cast  iron  

5000 

1 

5000 

1.78 

5000 
4 

5000 
16 

qtpol 

36000 
1 

36000 

1.78 

36000 
4 

36000 
16 

25000 

25000 

25000 

25000 

Carnegie's  hand-book  gives  S  =  50,000  for  medium  steel 
columns  and  q=  35000;  2^017  and  TSOOO  •  f°r  the  three  first 
columns  in  above  table. 

In  this  formula,  as  in  all  such,  the  values  of  the  constant 
should  be  determined  for  the  material  used  by  direct  experiment 
if  possible. 

W  I 

Or  use  straight  line  formula,  -r  =  S—k—  (12a) 


COLUMNS 


TABLE  la 

VALUES  OF  S  AND  k  IN  FORMULA  (12a) 
(Merriman's  Mechanics  of  Materials) 


Kind  of  column 

S 

k 

Limit  - 

Wrought  Iron: 
Flat  ends                               .    . 

42,000 

128 

218 

Hinged  ends  

42,000 

157 

178 

Round  ends         

42,000 

203 

138 

Mild  Steel: 
Flat  ends  

52,500 

179 

195 

Hinged  ends                

52,500 

220 

159 

Round  ends 

52,500 

284 

123 

Cast  Iron: 
Flat  ends                       

80,000 

438 

122 

Hinged  ends  
Round  ends  
Oak: 
Flat  ends  

80,000 
80,000 

5,400 

537 
693 

28 

99 

77 

128 

Carnegie's   hand-book   gives   allowable   stress  for   structural 
columns  of  mild  steel  as  12,000  for  lengths  less  than  90  radii,  and 

as  17,100  —57  -  for  longer  columns. 

This  allows  a  factor  of  safety  of  about  four. 


MACHINE  DESIGN 


TABLE  II 

CONSTANTS  OF  CROSS-SECTION 


Form  of 
section  and 
area  A 

Square  of 
radius  of 
gyration 

Moment 
of 
inertia 

Section 
modulus 

y 

Polar 
moment 
of  inertia 
J 

Torsion 
modulus 
J 

y 

Rectangle 

V 

bh3 

bh* 

W+Vk 

bh3+  b3h 

bh 

12 

12 

6 

12 

6^6'+  h* 

Square 

d2 

d4 

d3 

d4 

d3 

d2 

12 

12 

6 

6 

4.24 

Hollow 

bh3  —  61  h3i 

bh3  —  bih3i 

bh3  —  blh3i 

/-beam 

12(bh-bihi) 

12 

6h 

bh-bihi 

Circle 

d2 

;rd4 

d3 

;rd4 

d« 

16 

64 

10.2 

32 

5.1 

If 

Hollow 

d4-d*i 

d2+d2i 

3i(d4  —  d4i) 

^4_(f41 

re(d4  —  d4i) 

5.1d 

16 

64 

10.2d 

32 

Ellipse 

it 

a2 

xba3 

6«2 

x(ba3+  ab3) 

ba3+ab3 

16 

64 

10.2 

64 

10.2a 

Values  of  7  and  J  for  more  complicated  sections  can  be  worked  out  from 
those  in  table. 


LOADED  BEAMS 


TABLE  III 

FORMULAS  FOR  LOADED  BEAMS 


Beams  of  uniform  cross-section 

Maximum 
moment 
M 

Maximum 
deflection 

A 

Wl 

Wl3 

Cantilever  uniform  load 

Wl 

3EI 

Wl3 

Simple  beam  load  at  middle 

2 

Wl 

8EI 
Wl3 

Simple  beam   uniform  load 

4 
Wl 

±8EI 
5W13 

Beam  fixed  at  one  end,  supported  at  other, 

8 
3WI 

384EI 
.Q182W13 

load  at  middle. 
Beam  fixed  at  one  end,  suppported  at  other, 

16 

Wl 

El 

.0054  Wl3 

uniform  load. 
Beam  fixed  at  both  ends  load  at  middle 

8 
Wl 

El 

Wl3 

Beam  fixed  at  both  ends,  uniform  load  
Beam  fixed  at  both  ends,  load  at  one  end, 

8 

Wl 
^2 

Wl 

I92EI 

Wl3 
384EI 

Wl3 

(pulley  arm). 

2 

12EI 

5.  Profiles  of  Uniform  Strength. — In  a  bracket  or  beam  of 
uniform  cross-section  the  stress  on  the  outer  row  of  fibers  in- 
creases as  the  bending  moment  increases  and  the  piece  is  most 
liable  to  break  at  the  point  where  the  moment  is  a  maximum. 
This  difficulty  can  be  remedied  by  varying  the  cross-section  in 
such  a  way  as  to  keep  the  fiber  stress  constant  along  the  top  or 
bottom  of  the  piece.  The  following  table  shows  the  shapes  to 
be  used  under  different  conditions. 


MACHINE  DESIGN 


Type 

Load 

Plan 

Elevation 

Cantilever  
Cantilever.  .    . 

Center.  .  .  . 
Uniform   . 

Rectangle  .  .  . 
Rectangle  .  .  . 

Parabola,  axis  horizontal. 
Triangle. 

Simp.  Beam 

Center 

Rectangle 

Two  parabolas  intersecting 

Simp,  beam  

Uniform.  . 

Rectangle  .  .  . 

under  load. 
Ellipse,    major    axis  hori- 
zontal. 

The  material  is  best  economized  by  maintaining  a  constant 
breadth  and  varying  the  depth  as  indicated. 

This  method  of  design  is  applicable  to  cast  pieces  rather  than 
to  those  that  are  forged  or  cut. 

The  maximum  deflection  of  cantilevers  and  beams  having  a 
profile  of  uniform  strength  is  greater  than  when  the  cross-section 
is  uniform,  50  per  cent  greater  if  the  breadth  varies,  and  100 
per  cent  greater  if  the  depth  varies. 


CHAPTER  I 


MATERIALS 

6.  Primary  Classification. — The  materials  used  in  machine 
construction  are  practically  all  metals.  They  may  be  classified 
in  two  ways :  (a)  According  to  the  principal  metallic  constituents 
such  as  iron,  copper,  tin,  etc.;  (6)  as  cast  or  wrought  metals 
according  to  the  methods  employed  in  preparing  them  for  use. 

The  following  table  combines  the  two  methods  of  classification. 

TABLE  IV 


Principal  metal 

Cast 

Wrought 

Cast  iron 

Wrought  iron. 

Malleable  iron 

Soft  steel 

Copper  < 

Steel  castings  

Bronze  
Brass 

Tool  steel. 
Alloy  steel. 
Brass  wire. 
Sheet  brass. 

Tin  
Aluminum  

Babbitt  metal 
Bronze  

Rolled  or  drawn. 

7.  Iron. — Commercial  iron  is  produced  from  iron  ore  by 
reduction  in  a  blast  furnace.  Most  iron  ores  are  oxides  and  also 
contain  earthy  impurities  such  as  silica  and  alumina. 

The  oxygen  is  removed  by  the  burning  of  the  coke  used  as 
fuel,  while  the  limestone  used  as  a  flux  unites  with  the  silica 
and  alumina  forming  a  glassy  slag  which  floats  on  the  molten 
iron. 

Pig  Iron. — The  coarse-grained  impure  iron  thus  formed  is 
the  pig  iron  of  commerce  and  from  it  is  made  ordinary  cast  iron 
by  remelting  in  the  cupola  of  the  foundry.  Pig  iron  contains 
besides  iron  various  quantities  of  carbon,  silicon,  manganese, 
phosphorus  and  sulphur.  The  last  two  are  impurities  and  if 


10  MACHINE  DESIGN 

present  in  any  considerable  quantity  render  the  pig  unsuitable 
for  the  manufacture  of  high-grade  irons  or  steels.  The  phos- 
phorus comes  from  the  ore  and  the  sulphur  from  the  fuel  used. 
The  use  of  high-grade  ore  and  of  coke  made  from  a  non-sulphur 
coal  is  necessary  to  the  production  of  pure  iron.  Pig  iron  may 
be  used  in  the  foundry  for  the  manufacture  of  iron  castings,  in 
the  puddling  mill  for  producing  wrought  iron,  or  in  the  steel 
mill  for  the  manufacture  of  Bessemer  or  of  open-hearth  steel. 

Cast  Iron. — Iron  castings  are  made  in  the  foundry  by  melting 
pig  iron  in  a  cupola  using  coke  for  a  fuel.  The  quality  of  the 
cast  iron  depends  largely  upon  the  character  of  the  pig  iron  used, 
as  there  is  little  chemical  change  affected  in  the  cupola.  A 
certain  amount  of  scrap  cast  iron  may  be  melted  with  the 
charge;  remelting  of  iron  makes  it  finer  grained  and  harder. 
Wrought  iron  or  steel  shavings  mixed  with  the  molten  cast  iron 
produces  a  tough  fine-grained  iron,  sometimes  called  semi-steel. 
The  addition  of  about  25  per  cent  of  steel  scrap  makes  a  fine- 
grained soft  iron  having  a  tensile  strength  about  50  per  cent 
greater  than  that  of  the  cast  iron  without  the  steel. 

Carbon  exists  in  cast  iron  in  two  forms:  (a)  chemically  com- 
bined with  the  iron;  (6)  as  free  carbon  or  graphite.  The  larger 
the  proportion  of  free  carbon,  the  softer  and  weaker  is  the  iron. 
Remelting  and  cooling  increases  the  amount  of  combined  carbon 
and  makes  the  iron  harder  as  before  noticed.  The  total  amount 
of  carbon  present  varies  from  2  to  5  per  cent  in  different  irons. 

Silicon  is  an  important  element  in  iron  and  influences  the  rate 
of  cooling.  The  more  slowly  iron  cools  after  melting,  the  more 
graphite  forms,  the  less  the  shrinkage  and  the  softer  the  iron. 
Two  per  cent  of  silicon  gives  a  soft  gray  iron  with  a  high  tensile 
strength.  Machinery  iron  contains  usually  from  1^  to  2  per 
cent  of  silicon. 

Chilled  iron  is  cast  iron  which  has  been  cooled  suddenly  in  the 
mold  by  contact  with  metal  or  some  other  good  conductor  of 
heat.  Chilling  increases  the  amount  of  combined  carbon  and 
makes  the  iron  white  and  hard.  It  is  used  on  surfaces  which 
need  to  be  extremely  hard  and  durable,  as  the  treads  of  car 
wheels  and  the  outside  of  the  rolls  used  on  steel  mills.  The 
depth  of  the  chill  depends  on  the  amount  of  metal  used  in  the 
cooling  surface  of  the  mold. 


CAST  IRON  11 

All  castings  are  chilled  slightly  on  the  surface.  An'examina- 
tion  of  a  freshly  fractured  casting  shows  whiter  and  finer-grained 
metal  around  the  edges  than  at  the  center.  For  this  reason, 
castings  having  considerable  surface  or  "skin"  in  proportion  to 
their  weight  are  relatively  stronger  (see  Art.  15). 

In  selecting  cast  iron  for  various  machine  members,  soft  gray 
irons  should  be  chosen  where  workability  rather  than  strength  is 
desired.  Medium  gray  irons  having  a  fine  grain  should  be  used 
where  moderate  strength  and  hardness  are  necessary  as  in  the 
cylinders  of  steam  engines  and  pumps.  Hard  gray  iron  is  only 
suitable  for  heavy  castings  which  require  little  or  no  machining, 
as  it  is  brittle  and  not  easily  worked.  An  examination  of  the 
fracture  of  a  sample  of  iron  is  a  guide  in  determining  its  desira- 
bility for  any  particular  case. 

Cast  iron  is  the  cheapest  and  best  material  for  pieces  of  irregu- 
lar and  complicated  shape;  it  has  a  high  compressive  and  a  low 
tensile  strength;  it  is  brittle  and  cannot  be  welded  or  forged; 
but  it  resists  corrosion  much  better  than  wrought  iron.  For  its 
use  in  machine  construction,  see  Art.  14. 

Malleable  Iron. — Malleable  iron  is  cast  iron  annealed  and 
partially  decarbonized  by  being  heated  in  an  annealing  oven 
in  contact  with  some  oxidizing  material  such  as  hematite  ore, 
and  then  being  allowed  to  cool  slowly.  A  white  cast  iron  is 
best  for  this  process  as  the  presence  of  graphitic  carbon  interferes 
with  its  success.  An  iron  containing  a  small  amount  of  silicon 
and  considerable  manganese  promotes  the  formation  of  combined 
carbon  just  as  silicon  promotes  the  formation  of  free  carbon. 

The  castings  before  being  annealed  are  hard  and  brittle,  the 
fracture  showing  a  silvery  appearance.  They  are  packed  in 
air-tight  cast-iron  boxes  with  the  oxidizing  material  and  are 
kept  at  a  red  heat  for  several  days.  They  are  allowed  to  cool 
slowly  and  when  removed  are  tough  and  ductile  with  a  dull 
gray  fracture. 

The  oxidation  removes  some  of  the  total  carbon  from  the 
surface  of  the  material  and  the  heating  and  slow  cooling  changes 
the  most  of  that  remaining  to  graphite. 

An  iron  which  originally  contains  2.8  per  cent  combined  and 
0.20  per  cent  free  carbon,  after  annealing  may  show  0.20  per 
cent  combined  and  1.8  per  cent  free  carbon. 


12  MACHINE  DESIGN 

Malleable  castings  are  particularly  suitable  for  small  parts 
having  irregular  shapes.  The  metal  does  not  possess  as  much 
ductility  or  tensile  strength  as  wrought  iron  but  occupies  a  place 
intermediate  between  that  and  cast  iron. 

As  the  process  of  malleablizing  is  to  a  certain  extent  a  super- 
ficial one,  it  is  best  adapted  to  thin  metal,  although  castings  an 
inch  or  more  in  thickness  have  been  successfully  treated. 

Wrought  Iron. — Wrought  iron  is  commercially  pure  iron  which 
is  made  from  pig  iron  by  decarbonizing  it  in  the  puddling  furnace. 
This  furnace  is  a  reverberatory  one  in  which  the  molten  pig  is 
subjected  to  the  action  of  the  hot  gases  from  the  fuel. 

The  silicon,  manganese  and  carbon  are  oxidized  or  burned 
out,  either  by  the  action  of  the  gas  or  by  oxide  of  iron  introduced 
with  the  charge.  A  part  of  the  phosphorus  and  sulphur  is  also 
oxidized  in  the  puddling.  The  molten  mass  is  continually 
stirred  during  the  process  and  finally  assumes  a  pasty  consis- 
tency. It  is  then  squeezed  to  remove  the  slag  and  rolled 
into  bars.  These  are  cut,  piled  and  welded  into  either  bar  or 
plate  iron. 

The  particles  of  iron  in  the  puddling  process  are  more  or  less 
enveloped  in  the  slag  and  as  the  mass  is  squeezed  and  rolled, 
these  particles  become  fibers  separated  from  each  other  by  a  thin 
sheath  or  covering  of  slag,  and  it  is  this  which  gives  wrought 
iron  its  characteristic  structure. 

The  presence  of  either  sulphur  or  phosphorus  in  the  iron 
renders  it  less  reliable. 

Wrought  iron  possesses  moderate  tensile  strength  and  high 
ductility.  It  can  be  forged  and  welded  readily.  Hammering 
or  rolling  it  cold  increases  its  strength  and  stiffness  to  a  certain 
degree  and  raises  artificially  its  elastic  limit.  For  most  purposes, 
it  has  been  replaced  of  late  years  by  soft  steel.  Either  of  these 
metals  may  be  rendered  superficially  hard  by  the  process  known 
as  case  hardening.  The  pieces  to  be  treated  are  packed  in  air- 
tight boxes  together  with  pulverized  carbon  in  some  form, 
usually  bone-black.  The  boxes  are  brought  to  a  red  heat  and 
kept  so  for  several  hours.  The  pieces  are  then  removed  and 
quenched  suddenly  in  water.  The  surface  of  the  iron  has  com- 
bined with  the  carbon  in  which  it  was  packed  and  changed  to  a 
high-carbon  or  hardening  steel.  Such  pieces  have  a  soft,  ductile 


STEEL  13 

center  and  a  hard  surface.     Case  hardening  can  be  done  after 
finishing  but  is  liable  to  warp  the  metal. 

8.  Steel. — Steel  is  made  from  molten  pig  iron  by  burning  out 
the  silicon  and  carbon  with  a  hot  blast,  either  passing  through 
the  liquid  as  in  the  Bessemer  converter,  or  over  its  surface  as  in 
the  open-hearth  furnace.  A  suitable  quantity  of  carbon  and 
manganese  is  then  added  and  the  metal  poured  into  ingot  molds. 
If  the  ingots  are  reheated  and  rolled,  structural  steel  and  rods 
or  rails  are  the  result. 

Manganese  has  the  effect  of  preventing  blow  holes  and  giving 
the  steel  a  more  uniform  texture. 

Open-hearth  steel  differs  but  little  from  Bessemer  in  its  chemical 
composition  but  is  more  uniform  in  quality  on  account  of  the 
more  deliberate  nature  of  the  process  of  manufacture.  Boiler 
p'late,  structural  steel,  and  in  general  material  which  is  respon- 
sible for  the  safety  of  life  and  limb  should  be  of  open-hearth 
rather  than  Bessemer  steel. 

Steel  containing  not  more  than  0.6  per  cent  of  carbon  is  known 
as  soft  steel.  It  has  a  higher  elastic  limit  and  greater  tensile 
strength  than  wrought  iron,  which  metal  it  has  practically  sup- 
planted in  the  manufacture  of  machine  parts.  It  is  very  ductile 
and  malleable  and  may  be  welded  if  not  too  high  in  carbon. 

Crucible  steel  is  made  by  melting  steel  or  a  mixture  of  iron  and 
carbon  in  a  crucible  and  pouring  the  melted  metal  into  molds,  and 
hence  is  commonly  known  as  cast  steel. 

This  method  is  used  for  producing  the  harder  steels  suitable 
for  cutting  tools.  The  amount  of  carbon  will  vary  from  0.5  to 
1.5  per  cent  according  to  the  use  to  be  made  of  the  steel.  Such 
steel  contains  small  amounts  of  silicon  and  manganese  but  must 
be  practically  free  from  sulphur  and  phosphorus. 

It  is  relatively  high  priced  and  is  not  used  for  ordinary  machine 
parts.  It  cannot  be  readily  welded  but  possesses  the  very  useful 
characteristic  of  hardening  when  heated  to  a  red  heat  and  cooled 
suddenly.  The  degree  of  hardness  can  be  controlled  by  accu- 
rately measuring  the  temperature  of  heating  and  by  using  various 
cooling  agents  such  as  water,  brine  and  different  kinds  of  oil. 
The  steel  can  be  tempered  or  softened  after  hardening  by  reheat- 
ing to  a  slight  degree. 


14  MACHINE  DESIGN 

In  machine  construction  crucible  steel  is  only  used  for  screws, 
spindles,  ratchets,  etc.,  which  need  to  be  extremely  hard.  It 
has  a  high  tensile  and  compressive  strength  but  is  brittle  and 
liable  to  contain  hardening  cracks. 

Steel  castings  are  made  by  pouring  fluid  open-hearth  steel 
directly  into  molds.  They  possess  somewhat  the  same  charac- 
teristics as  malleable  castings,  being  relatively  tough  and 
ductile. 

It  has  been  somewhat  difficult  in  the  past  to  obtain  reliable 
castings  of  this  material  as  the  great  shrinkage — about  double 
that  of  cast  iron — has  tended  to  make  them  porous  and  spongy 
in  spots. 

Furthermore,  steel  which  was  sufficiently  low  in  carbon  to 
make  soft  castings  was  not  fluid  enough  to  run  sharply  in  the 
mold. 

These  difficulties  have  been  to  a  large  extent  overcome  and 
it  is  now  possible  to  obtain  steel  castings  which  are  reasonably 
clean  and  sound.  They  have  about  the  same  chemical  composi- 
tion as  mild  rolled  steel,  the  carbon  varying  from  0.2  to  0.6  per 
cent,  the  silicon  about  the  same  and  manganese  from  0.5  to  1 
per  cent.  Steel  castings  when  first  poured  are  coarse-grained 
and  should  be  annealed  to  make  them  tough  and  ductile. 

9.  Steel  Alloys. — Steel  alloys  are  compounds  of  steel  with 
chromium,  vanadium,  manganese,  etc.;  strictly  speaking,  all 
steels  are  alloys  of  iron  with  other  substances,  but  when  the  term 
steel  is  used  without  qualification,  it  is  understood  to  mean 
carbon  steel. 

Nickel  steel  is  both  stronger  and  tougher  than  carbon  steel. 
A  high  carbon  steel  is  strong  but  brittle;  the  same  or  greater 
strength  can  be  obtained  by  the  addition  of  nickel  without 
materially  diminishing  the  ductility.  This  metal  is  suitable  for 
pieces  which  are  subject  to  severe  shocks. 

Manganese  steel  is  an  alloy  containing  about  1  per  cent  of 
carbon  and  from  10  to  20  per  cent  of  manganese;  14  per  cent  of 
manganese  gives  the  maximum  of  strength  and  ductility  com- 
bined. This  metal  is  strong,  tough  and  extremely  hard,  so  that 
it  cannot  be  readily  finished  except  by  grinding.  It  can  be 
used  for  cutting  tools,  and  like  nickel  steel  is  valuable  for  pieces 


ALLOY  STEELS  15 

§ 

subjected  to  great  stress  and  wear.     Its  strength  is  increased  by 
heating  and  sudden  cooling. 

Chromium  is  sometimes  added  to  nickel  steel  in  the  manu- 
facture of  safes  and  armor  plate. 

Mushet  steel  is  an  alloy  of  high  carbon  steel  with  tungsten  and 
manganese  and  was  the  first  of  the  air-hardening  steels  used  for 
cutting  tools.  Like  all  of  this  class  of  tool  steels,  it  must  be 
worked  at  a  yellow  heat  and  hardens  when  cooled  slowly  in  the 
air. 

The  so-called  air-hardening  or  high-speed  steels  are  of  various 
chemical  compositions,  containing  carbon,  manganese,  tungsten, 
chromium,  molybdenum  or  titanium,  but  the  exact  ingredients 
and  proportions  are  for  the  most  part  trade  secrets.  Such 
steels  are  usually  purchased  in  small  sections  and  are  used  in 
special  tool  holders.  They  are  forged  with  great  difficulty  and 
are  generally  heated  in  special  furnaces  with  pyrometers  for 
determining  the  exact  temperature,  and  cooled  in  an  air  blast 
or  by  dipping  in  oil  baths.  The  difference  of  a  few  degrees  in  the 
temperature  of  the  metal  will  make  or  mar  the  cutting  efficiency. 
They  are  of  no  use  in  machine  construction,  but  affect  it  indirectly 
by  requiring  much  greater  strength,  rigidity  and  power  in 
machine  tools. 

It  is  not  an  uncommon  thing  for  the  power  consumption  of  a 
lathe  or  planer  to  be  increased  six  or  eight  times  by  the  use 
of  the  newer  tools. 

Vanadium  steel  is  one  of  the  latest  claimants  for  favor  among 
the  steel  alloys.  The  addition  of  a  small  amount  of  this  metal, 
0.1  or  0.2  per  cent,  increases  the  strength  and  stiffness  of  mild 
steel  in  a  marked  degree  with  comparatively  little  increase  in 
its  cost. 

It  is  already  used  extensively  in  machine  construction, 
particularly  in  marine  work. 

10.  Copper  Alloys. — These  metals  are  alloys  of  copper  and  tin, 
copper  and  zinc  or  of  all  three.  Copper  is  not  used  alone  in 
machine  construction  except  for  electric  conductors.  Phos- 
phorus, aluminum  and  manganese  are  also  used  in  combination 
with  copper. 

The  copper-tin  alloys  are  commonly  known  as  bronzes  and  are 


16  MACHINE  DESIGN 

expensive  on  account  of  the  large  proportion  of  copper,  from 
85  to  90  per  cent. 

Copper-zinc  alloys,  on  the  other  hand,  are  called  brass,  and  for 
maximum  strength  and  ductility  should  contain  from  60  to  70 
per  cent  of  copper. 

Bronzes  high  in  tin  and  low  in  copper  are  weak,  but  have 
considerable  ductility  and  make  good  metals  for  bearings. 
Tin  80,  copper  10  and  antimony  10  is  Babbitt  metal,  so  much 
used  to  line  journal  bearings,  the  antimony  increasing  the 
hardness. 

The  late  Dr.  Thurston's  experiments  on  the  copper-tin-zinc 
alloys  showed  a  maximum  strength  for  copper  55,  zinc  43  and 
tin  2  per  cent.  The  tensile  strength  of  this  mixture  was  nearly 
70,000  Ib.  per  square  inch. 

Phosphor  bronze  is  a  copper  alloy  with  a  small  amount  of 
phosphorus  added  to  prevent  oxidation  of  the  copper  and  thereby 
strengthen  the  alloy. 

Manganese  bronze  is  an  alloy  of  copper  and  manganese,  usually 
containing  iron  and  sometimes  tin.  A  bronze  containing  about 
84  per  cent  copper,  14  per  cent  manganese  and  a  little  iron, 
has  much  the  same  physical  characteristics  as  soft  steel  and 
resists  corrosion  better. 

There  is  practically  no  limit  to  the  varieties  of  color,  hardness, 
ductility  and"  durability  among  the  copper  alloys.  Some  of  the 
more  common  mixtures  are  here  given. 

TABLE  V 

COMPOSITION  OP  BRONZES   ' 


Name 


Composition 


Gun  metal 

Bell  metal 

Yellow  brass 

Muntz  metal 

Aluminum  bronze 

Phosphor  bronze 

Manganese  bronze  (1) . 

Manganese  bronze  (2) . 


Copper  .90,  tin  .10 
Copper  .77,  tin  .23 
Copper  .65,  zinc  .35 
Copper  .60,  zinc  .40 
Copper  .90,  aluminum  .10 
Copper  .89,  tin  .09,  phosphorus  .01 
Copper  .84,  manganese  .14,  iron  .02 
[Copper  .675,  manganese  .18 
\Zinc  .13,  aluminum  .01,  silicon  .005 


FACTORS  OF  SAFETY  17 

11.  Strength  and  Elasticity. — The  constants  for  strength  and 
elasticity  given  in  the  tables  are  only  fair  average  values  and 
should  be  determined  for  any  special  material  by  direct  experi- 
ment when  it  is  practicable.  Many  of  the  constants  are  not 
given  in  the  table  on  account  of  the  lack  of  reliable  data  for  their 
determination. 

The  strength  of  steel,  either  rolled  or  cast,  depends  so  much 
upon  the  percentages  of  carbon,  phosphorus  and  manganese, 
that  any  general  figures  are  liable  to  be  misleading.  Structural 
steel  usually  has  a  tensile  strength  of  about  65,000  Ib.  per  square 
inch,  while  boiler  plate  usually  has  less  carbon,  a  low  tensile 
strength  and  good  ductility. 

Factors  of  Safety. — A  factor  of  safety  is  the  ratio  of  the  ultimate 
strength  of  any  member  to  the  ordinary  working  load  which 
will  come  upon  it.  This  factor  is  intended  to  allow  for:  (a) 
Overloading  either  intentional  or  accidental,  (b)  Sudden  blows 
or  shocks,  (c)  Gradual  fatigue  or  deterioration  of  material. 
(d)  Flaws  or  imperfections  in  the  material. 

To  a  certain  extent  the  term  "factor  of  ignorance"  is  justifiable 
since  allowance  is  made  for  the  unknown.  Certain  fixed  laws 
may  guide  one,  however,  in  making  the  selection  of  a  factor. 
It  is  a  well-known  fact  that  loads  in  excess  of  the  elastic  limit  are 
liable  to  cause  failure  in  time.  Therefore,  when  the  elastic 
limit  of  the  material  can  be  determined,  it  should  be  used  as  a 
basis  rather  than  to  use  the  ultimate  strength. 

Furthermore,  suddenly  applied  loads  will  cause  about  double 
the  stress  due  to  dead  loads.  These  considerations  indicate  four 
as  the  least  factor  that  should  be  used  when  the  ultimate  strength 
is  taken  as  a  basis.  Pieces  subject  to  stress  alternately  in  oppo- 
site directions  should  have  large  factors  of  safety. 

The  following  table  shows  the  factors  used  in  good  practice 
under  various  conditions: 

Structural  steel  in  buildings 4 

Structural  steel  in  bridges 5 

Steel  in  machine  construction 6 

Steel  in  engine  construction 10 

Steel  plate  in  boilers 5 

Cast  iron  in  machines 6  to  15 

Castings  of  bronze  or  steel  should  have  larger  factors  than 
rolled  or  forged  metal  on  account  of  the  possibility  of  flaws. 


18 


MACHINE  DESIGN 


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20  MACHINE  DESIGN 

Cast  iron  should  not  be  used  in  pieces  subject  to  tension  or 
bending  if  there  is  a  liability  of  shocks  or  blows  coming  on  the 
piece. 

NOTE. — In  giving  references  to  transactions  and  periodicals,  the  follow- 
ing abbreviations  will  be  used: 

Transactions  of  American  Society  of  Mechanical  1  m          ~    ,_    „ 
-n      .  >  ir.  A.  fe.  M.  H/. 

Engineers.  J 

American  Machinist  Am.  Mach. 

Cassier's  Magazine  Cass. 

Engineering  Magazine  Eng.  Mag. 

Engineering  News  Eng.  News 

Machinery  Mchy. 

REFERENCES 

Materials  of  Machines.     A.  W.  Smith. 
Mechanics  of  Materials.     Merriman. 
Materials  of  Engineering.     Thurston. 


CHAPTER  II 
FRAME  DESIGN 

12.  General  Principles  of  Design. — The  working  or  moving 
parts  should  be  designed  first  and  the  frame  adapted  to  them. 

The  moving  parts  can  be  first  arranged  to  give  the  motions 
and  velocities  desired,  special  attention  being  paid  to  compact- 
ness and  to  the  convenience  of  the  operator. 

Novel  and  complicated  mechanisms  should  be  avoided  and 
the  more  simple  and  well-tried  devices  used. 

Any  device  which  is  new  should  be  first  tried  in  a  working 
model  before  being  introduced  in  the  design. 

The  dimensions  of  the  working  parts  for  strength  and  stiffness 
must  next  be  determined  and  the  design  for  the  frame  completed. 
This  may  involve  some  modification  of  the  moving  parts. 

In  designing  any  part  of  the  machine,  the  metal  must  be  put 
in  the  line  of  stress  and  bending  avoided  as  far  as  possible.  . 

Straight  lines  should  be  used  for  the  outlines  of  pieces  exposed 
to  tension  or  compression,  circular  cross-sections  for  all  parts  in 
torsion,  and  profile  of  uniform  fiber  stress  for  pieces  subjected 
to  bending  action. 

Superfluous  metal  must  be  avoided  and  this  excludes  all 
ornamentation  as  such.  There  should  be  a  good  practical 
reason  for  every  pound  of  metal  in  the  machine. 

An  excess  of  metal  is  sometimes  needed  to  give  inertia  and 
solidity  and  prevent  vibration,  as  in  the  frames  of  machines 
having  reciprocating  parts,  like  engines,  planers,  slotting  ma- 
chines, etc. 

Mr.  Oberlin  Smith  has  characterized  this  as  the  " anvil" 
style  of  design  in  contradistinction  to  the  " fiddle"  style. 

In  one  the  designer  relies  on  the  mass  of  the  metal,  in  the 
other  on  the  distribution  of  the  metal,  to  resist  the  applied  forces. 

A  comparison  of  the  massive  Tangye  bed  of  some  large  high- 
speed engines  with  the  comparatively  slight  girder  frame  used 
in  most  Corliss  engines,  will  emphasize  this  difference. 

21 


22  MACHINE  DESIGN 

It  may  be  sometimes  necessary  to  waste  metal  in  order  to 
save  labor  in  finishing,  and  in  general  the  aim  should  be  to 
economize  labor  rather  than  stock. 

The  designers  should  be  familiar  with  all  the  shop  processes 
as  well  as  the  principles  of  strength  and  stability.  The  usual 
tendency  in  design,  especially  of  cast-iron  work,  is  toward 
unnecessary  weight. 

All  corners  should  be  rounded  for  the  comfort  and  convenience 
of  the  operator,  no  cracks  or  sharp  internal  angles  left  where 
dirt  and  grease  may  accumulate,  and  in  general  special  attention 
should  be  paid  to  so  designing  the  machine  that  it  may  be  safely 
and  conveniently  operated,  that  it  may  be  easily  kept  clean, 
and  that  oil  holes  are  readily  accessible.  The  appearance  of  a 
machine  in  use  is  a  key  to  its  working  condition. 

Polished  metal  should  be  avoided  on  account  of  its  tendency 
to  rust,  and  neither  varnish  nor  bright  colors  tolerated.  The 
paint  should  be  of  some  neutral  tint  and  have  a  dead  finish  so 
as  not  to  show  scratches  or  dirt. 

Beauty  is  an  element  of  machine  design,  but  it  can  only  be 
attained  by  legitimate  means  which  are  appropriate  to  the 
material  and  the  surroundings. 

Beauty  is  a  natural  result  of  correct  mechanical  construction 
but  should  never  be  made  the  object  of  design. 

Harmony  of  design  may  be  secured  by  adopting  one  type  of 
cross-section  and  adhering  to  it  throughout,  never  combining 
cored  or  box  sections  with  ribbed  sections.  In  cast  pieces  the 
thickness  of  metal  should  be  uniform  to  avoid  cooling  strains, 
and  for  the  same  reason  sharp  corners  should  be  absent.  The 
lines  of  crystallization  in  castings  are  normal  to  the  cooled  surface 
and  where  two  flat  pieces  come  together  at  right  angles,  the 
interference  of  the  two  sets  of  crystals  forms  a  plane  of  weakness 
at  the  corner.  This  is  best  obviated  by  joining  the  two  planes 
with  a  bend  or  sweep. 

Rounding  the  external  corner  and  filleting  the  internal  one 
is  usually  sufficient.  Where  two  parts  come  together  in  such  a 
way  as  to  cause  an  increase  of  thickness  of  the  metal  there  are 
apt  to  be  "blow  holes"  or  "hot  spots"  at  the  junction  due  to 
the  uneven  cooling. 

"Strengthening"  flanges  when  of  improper  proportions  or  in 


FRAMES  23 

* 

the  wrong  location  are  frequently  a  source  of  weakness  rather 
than  strength.  A  cast  rib  or  flange  on  the  tension  side  of  a 
plate  exposed  to  bending,  will  sometimes  cause  rupture  by  crack- 


FIG.  1. — OLD    PLANING    MACHINE.     AN    EXAMPLE    OF    ELABORATE 
ORNAMENTATION. 

ing  on  the  outer  edge.     When  a  crack  is  once  started  rupture 
follows    almost    immediately.     When    apertures    are    cut    in    a 


24 


MACHINE  DESIGN 


frame  either  for  core-prints  or  for  lightness,  the  hole  or  aperture 
hsould  be  the  symmetrical  figure,  and  not  the  metal  that  sur- 
rounds it,  to  make  the  design  pleasing  to  the  eye. 

The  design  should  be  in  harmony  with  the  material  used  and 
not  imitation.  For  example,  to  imitate  structural  work  either 
of  wood  or  iron  in  a  cast-iron  frame  is  silly  and  meaningless. 


Machine  design  has  been  a  process  of  evolution.  The  earlier 
types  of  machines  were  built  before  the  general  introduction  of 
cast-iron  frames  and  had  frames  made  of  wood  or  stone,  paneled, 
carved  and  decorated  as  in  cabinet  or  architectural  designs. 


FRAMES  25 

t 

When  cast-iron  frames  and  supports  were  first  introduced 
they  were  made  to  imitate  wood  and  stone  construction,  so  that 
in  the  earlier  forms  we  find  panels,  moldings,  gothic  traceries 
and  elaborate  decorations  of  vines,  fruit  and  flowers,  the  whole 
covered  with  contrasting  colors  of  paint  and  varnished  as 
carefully  as  a  piece  of  furniture  for  the  drawing-room.  Relics  of 
this  transition  period  in  machine  architecture  may  be  seen  in 
almost  every  shop.  One  man  has  gone  down  to  posterity  as 
actually  advertising  an  upright  drill  designed  in  pure  Tuscan. 

13.  Machine  Supports. — The  fewer  the  number  of  supports  the 
better.  Heavy  frames,  as  of  large  engines,  lathes,  planers,  etc., 
are  best  made  so  as  to  rest  directly  on  a  masonry  foundation. 
Short  frames  as  those  of  shapers,  screw  machines  and  milling 
machines,  should  have  one  support  of  the  cabinet  form.  The  use 
of  a  cabinet  at  one  end  and  legs  at  the  other  is  offensive  to  the  eye, 
being  inharmonious.  If  two  cabinets  are  used  provision  should 
be  made  for  a  cradle  or  pivot  at  one  end  to  prevent  twisting  of 
the  frame  by  an  uneven  foundation.  The  use  of  intermediate 
supports  is  always  to  be  condemned,  as  it  tends  to  make  the 
frame  conform  to  the  inequalities  of  the  floor  or  foundation  on 
what  has  been  aptly  termed  the  "  caterpillar  principle." 

A  distinction  must  be  made  between  cabinets  or  supports  which 
are  broad  at  the  base  and  intended  to  be  fastened  to  the  founda- 
tion, and  legs  similar  to  those  of  a  table  or  chair.  The  latter  are 
intended  to  simply  rest  on  the  floor,  should  be  firmly  fastened  to 
the  machine  and  should  be  larger  at  the  upper  end  where  the 
greatest  bending  moment  will  come. 

The  use  of  legs  instead  of  cabinets  is  an  assumption  that  the 
frame  is  stiff  enough  to  withstand  all  stresses  that  come  upon  it, 
unaided  by  the  foundation,  and  if  that  is  the  case  intermediate 
supports  are  unnecessary. 

Whether  legs  or  cabinets  are  best  adapted  to  a  certain  machine 
the  designer  must  determine  for  himself. 

Where  two  supports  or  pairs  of  legs  are  necessary  under  a 
frame,  it  is  best  to  have  them  set  a  certain  distance  from  the 
ends,  and  make  the  overhanging  part  of  the  frame  of  a  parabolic 
form,  as  this  divides  up  the  bending  moment  and  allows  less 
deflection  at  the  center.  Trussing  a  long  cast-iron  frame  with 


26  MACHINE  DESIGN 

iron  or  steel  rods  is  objectionable  on  account  of  the  difference  in 
expansion  of  the  two  metals  and  the  liability  of  the  tension  nuts 
being  tampered  with  by  workmen. 

The  sprawling  double  curved  leg  which  originated  in  the  time 
of  Louis  XIV  and  which  has  served  in  turn  for  chairs,  pianos, 
stoves  and  finally  for  engine  lathes  is  wrong  both  from  a  practical 
and  esthetic  standpoint.  It  is  incorrect  in  principle  and  is 
therefore  ugly. 

EXERCISE 

1.  Apply  the  foregoing  principles  in  making  a  written  criticism  of  some 
engine  or  machine  frame  and  its  supports. 

(a)  Girder  frame  of  engine. 

(b)  Tangye  bed  of  air  compressor. 

(c)  Bed,  uprights  and  supports  of  iron  planing  machine. 

(d)  Bed  and  supports  of  engine  lathe. 

(e)  Cabinet  of  shaping  or  milling  machine. 

(f)  Frame  of  upright  drill. 

14.  Machine  Frames. — Cast  iron  is  the  material  most  used  but 
steel  castings  are  now  becoming  common  in  situations  where 
the  stresses  are  unusually  great,  as  in  the  frames  of  presses, 
shears  and  rolls  for  shaping  steel. 

Cored  vs.  Rib  Sections. — Formerly  the  flanged  or  rib  section  was 
used  almost  exclusively,  as  but  a  few  castings  were  made  from 
each  pattern  and  the  cost  of  the  latter  was  a  considerable  item. 
Of  late  years  the  use  of  hollow  sections  has  become  more  common; 
the  patterns  are  more  durable  and  more  easily  molded  than  those 
having  many  projections  and  the  frames  when  finished  are  more 
pleasing  in  appearance. 

The  first  cost  of  a  pattern  for  hollow  work,  including  the  cost 
of  the  core-box,  is  sometimes  considerably  more  but  the  pattern 
is  less  likely  to  change  its  shape  and  in  these  days  of  many 
castings  from  one  pattern,  this  latter  point  is  of  more  importance. 
Finally,  it  may  be  said  that  hollow  sections  are  usually  stronger 
for  the  same  weight  of  metal  than  any  that  can  be  shaped  from 
webs  and  flanges. 

Resistance  to  Bending. — Most  machine  frames  are  exposed  to 
bending  in  one  or  two  directions.  If  the  section  is  to  be  ribbed 
it  should  be  of  the  form  shown  in  Fig.  3.  The  metal  being  of 


FRAMES 


27 


nearly  uniform  thickness  and  the  flange  which  is  in  tension 
having  an  area  three  or  four  times  that  of  the  compression  flange. 
In  a  steel  casting  these  may  be  more  nearly  equal.  The  hollow 
section  may  be  of  the  shape  shown  in  Fig.  4,  a  hollow  rectangle 
.with  the  tension  side  re-enforced  and  slightly  thicker  than  the 
other  three  sides.  The  re-enforcing  flanges  at  A  and  B  may  often 
be  utilized  for  the  attaching  of  other  members  to  the  frame  as  in 
shapers  or  drill  presses.  The  box  section  has  one  great  advantage 
over  the  I  section  in  that  its  moment  of  resistance  to  side  bending 


FIG.  3. 


FIG.  4. 


or  to  twisting  is  usually  much  greater.  The  double  I  or  the  U 
section  is  common  where  it  is  necessary  to  have  two  parallel 
ways  for  sliding  pieces  as  in  lathes  and  planers.  As  is  shown  in 
Fig.  5  the  two  Fs  are  usually  connected  at  intervals  by  cross 
girts. 

Besides  making  the  cross-section  of  the 
most  economical  form,  it  is  often  desirable 
to  have  such  a  longitudinal  profile  as  shall 
give  a  uniform  fiber  stress  from  end  to 
end.  This  necessitates  a  parabolic  or 
elliptic  outline  of  which  the  best  instance 
is  the  housing  or  upright  of  a  modern  iron 
planer. 

Resistance  to  Twisting. — 'The  hollow  cir- 
cular section  is  the  ideal  form  for  all  frames  or  machine  mem- 
bers which  are  subjected  to  torsion.  If  subjected  also  to  bend- 
ing the  section  may  be  made  elliptical  or,  as  is  more  common, 
thickened  on  two  sides  by  making  the  core  oval.  See  Fig.  6. 
As  has  already  been  pointed  out  the  box  sections  are  in  general 
better  adapted  to  resist  twisting  than  the  ribbed  or  I  sections. 


FIG.  5. 


28 


MACHINE  DESIGN 


FIG.  6. 


Frames  of  Machine  Tools.— The  beds  of  lathes  are  subjected 
to  bending  on  account  of  their  own  weight  and  that  of  the  saddle 
and  on  account  of  the  downward  pressure  on  the  tool  when  work 
is  being  turned.  They  are  usually  subjected  to  torsion  on  ac- 
count of  the  uneven  pressure  of  the  supports.  The  box  section 
is  then  the  best;  the  double  I  commonly 
used  is  very  weak  against  twisting.  The 
same  principle  would  apply  in  designing 
the  beds  of  planers  but  the  usual  method 
of  driving  the  table  by  means  of  a  gear 
and  rack  prevents  the  use  of  the  box  sec- 
tion. The  uprights  of  planers  and  the 
cross  rail  are  subjected  to  severe  bending 
moments  and  should  have  profiles  of  uni- 
form strength.  The  uprights  are  also  sub- 
ject to  side  bending  when  the  tool  is  taking  a  heavy  side  cut  near 
the  top.  To  provide  for  this  the  uprights  may  be  of  a  box  sec- 
tion or  may  be  reinforced  by  outside  ribs. 

The  upright  of  a  drill  press  or  vertical  shaper  is  exposed  to  a 
constant  bending  moment  equal  to  the  upward  pressure  on  the 
cutter  multiplied  by  the  distance 
from  center  of  cutter  to  center 
of  upright.  It  should  then  be 
of  constant  cross-section  from 
the  bottom  to  the  top  of  the 
straight  part.  The  curved  or 
goose-necked  portion  should 
then  taper  gradually. 

The  frame  of  a  shear  press  or 
punch  is  usually  of  the  G  shape 
in  profile  with  the  inner  fibers  in 
tension  and  the  outer  in  compression.  The  cross-section  should 
be  as  in  Fig.  3  or  Fig.  4,  preferably  the  latter,  and  should  be 
graduated  to  the  magnitude  of  the  bending  moment  at  each 
point.  (See  Fig.  7.) 

15.  Tests  on  Simple  Beams. — In  1902,  a  series  of  experiments 
was  made  on  cast-iron  beams  of  various  sections  at  the  Case 
School  of  Applied  Science.  The  work  was  done  by  Messrs. 


FIG.  7. 


CAST  IRON  BEAMS  29 

A.  F.  Kwis  and  R.  H.  West1  under  the  direction  of  the  author 
and  the  results  were  reported  by  him  in  1906. 

The  patterns  were  all  20  in.  long  and  had  the  same  cross-section 
of  4.15  sq.  in.  As  may  be  seen  from  the  tables,  the  areas  of  the 
cast  beams  varied  slightly.  The  castings  of  each  set  were  all 
made  from  the  same  ladle  of  iron  and  were  cast  on  end.  A  soft 
gray  iron  was  used  and  a.  large  flush  basin  distributed  the  molten 
metal  to  the  mold,  giving  a  uniform  temperature  and  quality. 
The  castings  were  prepared  by  Mr.  Thomas  D.  West  and  proved 
to  be  remarkably  uniform  in  quality  and  free  from  imperfections. 

The  specimens  were  all  tested  by  loading  transversely  at  the 
center,  the  supports  being  18  in.  apart. 

Object. — The  investigation  had  two  distinct  objects  in  view 
and  two  classes  of  test  pieces  were  used.  The  first  class  com- 
prised Nos.  1  to  11  and  Nos.  22  to  32,  and  these  specimens  had 
sections  such  as  are  used  in  parts  of  machines. 

The  second  class  comprised  Nos.  12  to  21  and  33  to  42,  all 
having  sections  similar  to  those  used  in  the  rims  of  fly-wheels. 
The  sections  tested  were  such  as  shown  by  the  diagrams  in  the 
tables. 

The  areas  given  in  the  table  are  those  of  the  specimens  at  the 
point  of  rupture.  There  are  two  specimens  of  each  shape  cast 
from  the  same  pattern. 

The  section  modulus  -  was    calculated  from  the  dimensions 

of  the  casting  at  the  breaking  point,  y  being  the  distance  from 
neutral  axis  to  extreme  fiber  in  tension.  In  testing  each  specimen 
the  load  was  applied  gradually  and  readings  of  the  deflection 
were  taken  at  regular  intervals.  When  the  "set"  load  was 
reached,  the  pressure  was  removed  and  a  reading  of  the  perma- 
nent set  was  taken.  The  load  was  again  applied  and  observations 
made  on  the  deflection  up  to  near  the  time  of  rupture. 

The  load-deflection  curves  plotted  from  these  observations 
are  nearly  all  smooth  and  uniform  in  character,  as  may  be  seen 
by  reference  to  Fig.  8  which  shows  the  curves  for  No.  33. 

The  initial  line  curves  gradually  from  the  start  showing  an 
imperfect  elasticity,  while  the  set  line  is  nearly  straight  and 
approximately  parallel  to  the  tangent  of  the  curve  at  the  vertex. 

1  Mchy.,  May,  1906. 


30 


MACHINE  DESIGN 


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CAST  IRON  BEAMS  31 

The  so-called  moduli  of  elasticity  were  calculated 'from  the 
set  lines  using  the  formula 

E 


48  A/ 

In  each  test  a  reading  of  the  load  was  taken  at  the  instant 
when  the  deflection  measured  0.03  in.,  and  these  loads  may  be 
taken  as  a  fair  measure  of  the  "stiffness"  of  the  section. 

The  modulus  of  rupture  was  calculated  from  the  breaking  load 
and  the  section  modulus,  using  the  formula: 

My     Wly 

~T'-  =  ^T 

The  modulus  of  rupture,  as  S  is  generally  called,  is  supposed  to 
represent  the  tensile  stress  on  the  outer  fibers  at  the  point  of 
rupture  and  to  measure  in  a  way  the  transverse  strength  of  the 
material.  In  the  absence  of  a  better  measure  we  will  use  this, 
and  take  the  circular  and  square  sections  as  our  standards.  The 
average  value  of  S  for  the  four  is  24,360  Ib.  per  square  inch. 

This  is  a  low  value  even  for  soft  gray  iron.  The  remarkable 
fluctuations  in  the  value  of  $  for  specimens  of  different  cross- 
section,  from  a  minimum  of  18,700  to  a  maximum  of  36,000, 
show  that  the  ordinary  method  of  calculation  would  not  be  of 
much  value  in  predicting  the  breaking  load  of  such  beams. 

Comparison  of  Strength. — An  investigation  of  the  values  in 
Table  VIII  shows  that  the  hollow  circular  and  elliptic  sections 
are  much  stronger  than  the  solid  sections,  the  increase  in  strength 
being  greater  than  that  of  the  section  modulus.  The  average  value 
of  S  for  the  last  six  numbers  in  Table  VIII  is  31,600  as  against 
24,000  for  the  six  solid  sections,  an  apparent  increase  in  the 
strength  of  the  material  itself  of  over  25  per  cent.  This  is  partly 
due  to  the  thinner  metal,  the  greater  surface  of  hard  "skin" 
and  the  freedom  from  shrinkage  strains. 

The  absence  of  corners  and  the  consequent  uniformity  of 
metal  make  this  an  ideal  form  of  section. 

The  hollow  rectangles  and  the  I-sections  given  in  Table  IX 
have  an  average  value  of  S  =  22,450. 

No.  8  is  lower  than  the  average  and  Nos.  28  and  32  considerably 
higher.  These  discrepancies  are  due  to  some  accidental  condi- 


32 


MACHINE  DESIGN 


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34  MACHINE  DESIGN 

tion  of  the  metal,  since  the  mates  of  these  pieces  had  about  the 
average  strength. 

The  relatively  low  values  of  S  for  this  series  are  probably  due 
to  cooling  strains  in  the  metal.  The  table  shows  quite  conclu- 
sively that  the  increase  in  strength  in  such  sections  is  not  pro- 
portional to  the  increase  in  the  section  modulus. 

Elasticity. — The  values  for  the  modulus  of  elasticity  in  Tables 
VIII  and  IX  seem  almost  ridiculous,  if  we  are  to  regard  this 
much  abused  " constant"  as  any  criterion  of  the  stiffness  of  a 
beam. 

According  to  the  results  of  tensile  and  transverse  tests  on 
cast  iron  E  is  a  variable,  being  greatest  for  small  loads  and 
diminishing  as  we  approach  the  breaking  load. 

Prof.  Lanza  gives  values  varying  from  nine  to  eighteen  millions 
for  a  test  on  one  bar.  As  has  been  explained,  the  values  of  E 
were  determined  from  the  set  lines  which  were  approximately 
straight  and  not  subject  to  the  variation  above  mentioned. 
Examining  the  tables  we  find  the  values  of  E  ranging  all  the  way 
from  11,000,000  down  to  3,290,000. 

The  larger  values  go  with  the  smaller  depths  as  in  Nos.  17  and 
38  and  the  smaller  values  are  found  in  the  sections  having  the 
largest  section  moduli  as  in  Nos.  7  to  11. 

1  This  goes  to  show  that  the  common  formula  for  E  does  not 
apply  well  in  the  case  of  cast-iron  sections  and  that  the  deflection 
of  hollow  and  I-shaped  sections  is  much  greater  than  would  be 
given  by  the  formula.  The  columns  giving  the  loads  for  a 
deflection  of  0.03  in.  illustrate  this.  For  instance,  the  values  of 
7  for  Nos.  1  and  32  are  1.545  and  12.67  respectively,  having  a 
ratio  of  8.2. 

The  loads  required  to  produce  the  same  deflection  of  0.03 
in.  are  2500  Ib.  and  13,300  Ib.  respectively,  having  a  ratio  of 
only  5.3. 

Rim  Sections. — The  object  of  the  experiments  summarized  in 
Tables  X  and  XI  was  to  determine  the  effect  of  flanges  on  the 
strength  and  stiffness  of  sections  such  as  are  used  for  the  rims 
of  fly-wheels. 

In  order  to  illustrate  this  more  clearly  each  alternate  section 
was  turned  over  so  as  to  bring  the  flanges  on  the  tension  side, 
as  may  be  seen  by  the  shapes  in  the  second  columns  of  the  tables. 


WHEEL  RIMS 


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36  MACHINE  DESIGN 

The  section  modulus  and  the  fiber  stress  were  always  calculated 
from  the  tension  side. 

In  nearly  every  instance  the  calculated  value  of  S  is  higher 
for  the  beam  having  the  web  in  compression  and  the  flanges 
in  tension,  or  in  other  words  there  is  not  so  much  disadvantage 
in  this  latter  arrangement  as  theory  would  indicate. 

For  instance,  the  section  modulus  for  No.  34  is  more  than 
twice  that  of  No.  13  of  similar  shape  and  area,  but  the  breaking 
load  is  only  one-third  greater.  If  we  knew  where  the  neutral 
axes  of  these  sections  really  were  during  the  process  of  bending 
we  might  perhaps  explain  this  discrepancy. 

Depth  of  Flanges. — Another  object  of  these  experiments  on 
wheel  rim  sections  was  to  determine  the  relative  value  of  shallow 
and  deep  flanges.  The  average  value  of  the  breaking  load  for 
the  ten  sections  with  shallow  flanges  in  Table  X  is  6690  lb., 
and  the  average  value  of  S  about  25,000  lb.  per  square  inch. 
The  corresponding  values  for  the  ten  sections  with  deeper  flanges 
in  Table  XI  are  11,800  and  22,800.  There  is  thus  a  slight  falling 
off  in  the  value  of  S  for  the  deeper  sections  but  not  so  much  as 
was  noticed  in  the  two  other  tables. 

The  elasticity  of  these  sections  is  more  uniform  than  in  those 
previously  noticed,  E  varying  from  six  to  eleven  millions.  We 
notice,  however,  the  same  peculiarity  as  before,  that  the  deeper 
sections  are  not  so  stiff  in  proportion  to  the  values  of  7  as  those 
having  shallow  flanges. 

The  conclusions  to  be  derived  from  these  experiments  can  be 
stated  in  a  few  words: 

(1)  The  commonly  accepted  formulas  for  the  strength  and 
stiffness  of  beams  do  not  apply  well  to  cored  and  ribbed  sections 
of  cast  iron. 

(2)  Neither  the  strength  nor  the  stiffness  of  a  section  increases 
in  proportion  to  the  increase  in  the  section  modulus  or  the 
moment  of  inertia. 

(3)  The  best  way  to  determine  these  qualities  for  a  cast-iron 
beam  is  by  experiment  with  the  particular  section  desired  and 
not  by  reasoning  from  any  other  section. 

The  experiments  described  in  this  article  were  made  with 
unusual  care  on  a  remarkably  clean  and  homogeneous  iron  and 
the  regularity  of  the  load  curves  shows  accurate  measurement. 


WHEEL  RIMS 


37 


That  the  calculated  stresses  and  moduli  show  so  wide  a  diver- 
gence must  be  attributed  to  the  formulas  rather  than  the  work 


No.  33 


7500  Ib. 


5000  Id. 


2500  Ib. 


.05 


JO 


.15 


FIG.  8. — LOAD  DEFLECTION  CURVES  FOR  SAMPLE  No.  33. 

A  set  of  preliminary  experiments  made  on  similar  sections  in 
1901  gave  results  almost  identical  with  those  described,  the  values 
of  S  ranging  from  22,000  to  35,000  and  those  of  E  from  five  to  nine 


38 


MACHINE  DESIGN 


millions  for  a  rather  hard  gray  iron.     The  hollow  circular  sections 
made  the  best  showing  and  the  thin,  deep  I-sections  the  poorest. 

16.  Shapes  of  Frames. — The  contours  or  outlines  of  machine 
frames  vary  with  the  work  to  be  done  and  the  degree  of  accessi- 
bility desired.  They  may  be  roughly  classified  as  follows: 

(a)  \-\  or  parallel  type,  with  symmetrical  loading  and  direct 
tension  or  compression  in  parallel  members. 

(b)  A  or  triangular  type,  with  direct  tension  or  compression  in 
inclined  members  and  also  in  cross  girt. 

(c)  £  or  eccentric  type  with  combined  tension  and  bending  in 
long   member,   bending   and   shear  in   two   parallel   members. 
Similar  to  the  column  with  eccentric  loading. 

(d)  C  type  similar  to  (c),  a  semi-circular  member  being  sub- 
stituted for  the  long  straight  member.     Variable  tension  and 
bending  combined  with  shear  throughout  curved  part. 

(e)  Q  or  open  circular  type  with  variable,  combined  stresses 
as  in  circular  part  of  (d). 

(/)  O  or  dosed  circular  type  with  combined  stresses  varying 
throughout. 

Numerous  combinations  of  these  various  elements  can  be 
designed  but  the  principles  will  remain  the  same.  Table  XII  is 
convenient  for  reference. 

TABLE  XII 


Stresses  in  members 

Type 

Illustration 

Vertical 

Horizontal 

(a)    H   .. 

Tension,  or  compres- 

Negligible   

Hydraulic  press,  slotting  ma- 

sion. 

chine. 

(6)    A  ... 

Tension,  or  compres- 

Compression, or  ten- 

Engine frames. 

sion. 

sion. 

(c)    E... 

Tension  and  Bending 

Bending  and  shear.  .  . 

Side-crank  engine,  drill  press. 

(d)    C  - 

Variable  
Combined  

Bending  and  shear.  .  . 

Punch  or  shear  frame. 

(e)    C  ... 

Variable  

Variable  1 

Crane  hook.  1 

Combined.                     /  j  C-nlamn.          1 

(/)    0... 

Variable. 

Variable  \ 
Combined  J 

Chain  link. 

Combined. 

NOTE. — The  load  is  assumed  to  be  vertical  in  each  case. 


FRAMES  39 

• 

17.  Stresses  in  Frames.  —  The  design  of  frames  of  the  first  two 
types  in  Table  XII  involves  no  serious  difficulty  as  the  stresses 

are  comparatively  simple.     The  ratio  -  is  usually  too  small  to 

permit  of  buckling  in  the  straight  members.  As  in  all  cast-iron 
work,  care  must  be  taken  in  proportioning  ribs  and  fillets  to  avoid 
serious  cooling  strains  and  allowance  must  be  made  for  the 
inferior  strength  of  large  castings  as  compared  with  small. 

When  we  consider  types  (c),  (d)  and  (e)  where  the  loading  is 
eccentric  and  the  stresses  are  composite,  the  problem  is  much 
less  easy  of  solution. 

Cast  iron  is  the  material  most  used  for  machine  frames  and 
cast  iron  is  not  perfectly  elastic.  The  stress-strain  diagram  is 
not  straight  but  parabolic  (see  Fig.  8)  and  presents  no  well- 
defined  elastic  limit. 

From  Hodgkinson's  experiments,  the  laws  governing  the 
relations  between  unit  stress  and  unit  deformation  were  found 
to  be  approximately  expressed  thus: 

For  tension  : 

S  =  1,  400,000s  (1-  209s) 
For  compression: 

S  =  1,  300,000s  (1-  40s) 
where 

/S  —  unit  stress 

s  =  unit  deformation. 


Since  the  material  does  not  obey  Hooke's  law,  the  ordinary 
formulas  for  beams  will  apply  only  within  narrow  limits.  The 
attempt  to  apply  the  more  complicated  formulas  of  Resal  and 
Andrews-Pearson  can  only  result  in  a  waste  of  time.  1 

Under  such  circumstances,  it  is  best  to  use  simple  formulas  and 
determine  the  constants  by  experiment  as  has  been  done  in  the 
case  of  columns  (see  art.  4). 

18.  Professor  Jenkin's  Experiments.  —  The  first,  experiments 
so  far  reported  which  throw  much  light  on  this  particular 
problem  are  those  made  by  Professor  A.  L.  Jenkins  and  reported 

1  For  a  discussion  of  these  formulas,  see  Slocum  and  Hancock's  Strength 
of  Materials,  Chapter  IX,  and  Proc.  A.  S.  M.  E.,  May,  1910. 


40 


MACHINE  DESIGN 


by  him  in  the  proceedings  of  the  American  Society  of  Mechanical 

Engineers.1 

»    The  castings  tested  by  him  were  eighteen  in  number  and  of 

three  different  forms,  all  being  models  on  a  reduced  scale  of 

ordinary  punch  or  riveter  frames  somewhat  similar  to  the  one 

shown  in  Fig.  7. 


FIG.  9. 


Fig.  9  shows  the  three  typical  forms  chosen:  (a)  Plain  section 
with  curved  throat;  (b)  ribbed  section  with  curved  throat;  (c) 
plain  section  with  straight  throat.  All  of  the  specimens  were 
small,  the  depth  of  gap  being  only  6  or  7  in. 


FIG.  10. 

Table  XIII  gives  the  most  important  results  of  the  experiments. 
The  stress  in  the  last  column  was  calculated  by  the  formula, 

rr     My      W  /IQ\ 

b=-j-  +  -£  (16) 

the  notation  being  the  same  as  is  used  elsewhere  in  this  book. 
1  Proc.  A.  S.  M.  E.,  May,  1910. 


TESTS  OF  FRAMES 


41 


TABLE  XIII 

JENKIN'S  EXPERIMENTS  ON  RIVETER  FRAMES 


Strength  of 

Strength  of 

test  bar 

frame  section 

Tensile 

Trans- 

Breaking 

Unit 

Remarks 

verse 

stress 

stress 

load 

stress 

(at  A) 

1 

19,100 

36,560 

11,200 

16,240  i 

2 

18,620 

44,200 

11,125 

16,120  1 

Same  as  (a),  Fig.  9. 

3 

19,000 

46,080 

11,390 

16,540  J 

4 
5 

21,630 
21,630 

37,200 
40,000 

9,300 
8,500 

11,330  i 
10,500  J 

Same  as  (6)  ,  Fig.  9. 

6 

18,600 

39,000 

12,600 

22,520 

(6)  with  web  thickened. 

7 
8 

18,750 
21,700 

43,000 
46,250 

12,000 
15,300 

9,790  i 
12,600  J 

Tested  in  compression. 

9 

22,920 

39,600 

8,300 

10,130 

(6)  with  outer  flange  reduced. 

10 

20,370 

43,700 

8,400 

10,520 

(6)  with  inner  flange  reduced. 

11 

23,600 

36,400 

5,200 

18,420 

(6)  with  both  flanges  reduced. 

12 

23,000 

38,000 

8,400 

10,235 

(6)  with  both  flanges  reduced. 

13 

24,400 

45,000 

5,800 



(6)  with  both  flanges  notched. 

14 

21,800 

40,600 

12,700 

23,920 

(6)  with  fillet  strengthened. 

15 

21,400 

40,400 

12,500 

23,400 

(6)  with  outer  flange  removed. 

16 
17 

21,270 
22,080 

37,800 
42,200 

11,255 
11,980 

16,320  \ 
17,270  / 

Same  as  (c),  Fig.  9. 

18 

22,800 

41,300 

10,600 

21,476 

Depth  of  spine  reduced. 

That  is,  the  tensile  strength  at  the  inside  flange  is  the  sum  of 
that  due  to  the  bending  moment  and  that  due  to  direct  tension. 

Some  of  the  different  lines  of  fracture  are  indicated  in  Fig.  10, 
the  number  of  each  line  corresponding  to  the  piece  number. 
Number  5  shows  an  apparent  weakness  in  the  web  near  the 
flange  probably  due  to  cooling  strains,  since  the  inner  flange  was 
thicker  than  the  web  (see  cylinder  flanges,  pp.  80  and  81). 
Thickening  the  web  as  in  No.  6  changed  this  and  increased  the 
strength  (see  Table  XIII).  When  a  box  section  is  employed, 
the  change  in  thickness  between  the  inner  plate  and  the  side  plate 
should  be  gradual. 

Removing  or  reducing  the  inner  flange  always  weakened  the 
piece  (compare  Nos.  6  and  10).  Removing  the  outer  flange  did 
not  always  affect  the  strength  (compare  Nos.  14  and  15).  The 
specimen  with  a  straight  throat  of  the  (c)  shape  usually  broke  in 


42  MACHINE  DESIGN 

the  round  corner  as  might  be  expected  from  the  nature  of  the 
material  (see  Art.  12). 

The  load-deflection  curves  obtained  in  these  tests  by  means 
of  an  autographic  recorder  are  similar  in  character  to  those 
obtained  by  the  author  from  cast-iron  beams  (see  Fig.  8)  and 
show  no  evidence  of  a  yield-point  or  an  elastic  limit.  The  con- 
clusions reached  by  Professor  Jenkins  as  a  result  of  these  tests  are 
here  given  verbatim. 

"Although  these  experiments  are  not  sufficiently  exhaustive 
to  render  any  rigid  conclusions,  they  seem  to  indicate  that  the 
following  statements  are  approximately  true: 

(a)  There  is  no  rational  method  for  predicting  the  strength  of  curved 
cast-iron  beams  suitable  for  punch  and  shear  frames. 

(b)  Of  the  three  formulas  suggested  for  the  design  of  punch  frames,  the 
well-known  beam  formula, 


is  the  most  accurate  statement  of  the  law  of  stress  relations  existing 
in  such  specimens. 

(c)  The  stress  behind  the  inner  flange  at  the  curved  portion  is  an  impor- 
tant consideration  that  should  be  recognized  by  the  designer. 

(d)  There  seems  to  be  no  definite  relation  existing  between  the  strength  of 
a  curved  cast-iron  beam  and  the  transverse  strength  of  a  test  bar  cast 
with  it. 

(e)  The  Rcsal  and  Pearson-Andrews  formulas  are  unwieldy  and  awkward 
in  their  application  and  offer  many  chances  for  error." 

The  somewhat  erratic  variations  of  the  value  of  the  calculated 
unit  stresses  in  Table  XIII  are  rather  discouraging  to  the  designer 
but  are  really  no  worse  than  those  obtained  from  simple  beams, 
as  may  be  seen  by  reference  to  Tables  VIII  to  XI  inclusive. 

19.  Purdue  Tests.  —  During  the  past  year  some  experiments  on 
curved  frames  were  conducted  by  Messrs.  Charters,  Harter  and 
Luhn  of  the  senior  class  in  the  Testing  Materials  Laboratory  of 
Purdue  University. 

The  characteristic  shape  and  dimensions  of  the  specimens  are 
indicated  in  Fig.  11,  while  Fig.  12  shows  the  piece  in  position  in 
the  testing  machine.  The  load  was  applied  by  means  of  stirrups 
carrying  round  steel  pins  which  bore  in  the  milled  grooves  shown 
at  G,  Fig.  11. 


RIVETER  FRAMES  43 

The  proportions  of  the  frame  were  copied  from  those  of  a  large 
hydraulic  riveter  made  by  a  reputable  firm.  The  castings  were 
of  a  uniform  quality  of  soft  gray  iron  and  were  made  in  the 
university  foundry. 

Test  pieces  for  tension  and  flexure  were  cast  from  the  same  heat 
and  showed  an  average  tensile  strength  of  about  25,000  Ib.  and  a 
modulus  of  rupture  of  about  41,500  Ib. 

Twenty-four  pieces  were  broken,  the  same  pattern  being  used 
throughout,  various  modifications  being  made  in  the  flanges  and 
fillets. 


FIG.    11. 

The  following  table  shows  these  modifications  in  detail  and  the 
effect  which  they  had  on  the  strength  and  stiffness  of  the  frames. 
Some  of  the  characteristic  lines  of  fracture  are  shown  in  Fig.  11, 
each  line  being  numbered  to  correspond  to  the  number  of  the 
specimen. 

The  first  twelve  specimens  broke  by  splitting  the  web  along  a 
curved  line  parallel  to  and  adjacent  to  the  inner  flange.  This 
type  of  break  has  already  been  discussed  in  Art.  18. 

Numbers  13  to  16,  inclusive,  broke  directly  across  the  frames 
in  lines  parallel  to  one  of  the  radial  ribs. 

Numbers  17  and  18  broke  in  much  the  same  manner  in  lines 
parallel  to  the  one  rib. 

Numbers  19  and  20  started  a  fracture  in  the  web  adjacent  to 
the  rib  but  this  did  not  extend  through  the  flanges. 

The  last  four  frames  broke  in  a  practically  vertical  line  through 
web  and  flanges  just  back  of  the  inner  flanges. 


44 


MACHINE  DESIGN 


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45 


A  study  of  the  values  given  in  the  table  and  of  the  lines  of 
fracture  in  Fig.  11  shows  the  difficulty  of  applying  any  general 
formula  to  this  problem. 

The  general  tendency  of  all  the  frames  which  are  not  reinforced 
by  radial  ribs  is  to  split  or  shear  in  the  web.  Probably  all  of 


FIG.    12. 


the  fractures  begin  in  this  way  and  it  is  more  or  less  a  matter  of 
chance  whether  the  fracture  extends  through  the  flanges. 

When  ribs  are  used,  the  tendency  is  still  to  shear  the  web 
alongside  of  the  rib.  as  in  16  and  18,  with  a  possibility  of  the  break 
not  extending  through  the  flanges  (see  No.  20). 

It  is  apparent  that  thickening  the  flanges  will  do  no  good, 
while  thickening  the  web  is  efficient  (see  Nos.  21-22).  Changing 
the  thickness  of  web  from  f  in.  to  0.6  in.  increased  both  strength 
and  stiffness  nearly  100  per  cent. 


46 


MACHINE  DESIGN 


DESIGN  OF  FRAMES  47 

The  addition  of  J-in.  fillets  increased  strength  and  stiffness, 
while  ^-in.  fillets  were  less  effective. 

Although  these  experiments  were  not  sufficient  in  number  to 
justify  definite  conclusions,  it  is  evident  that  the  web  and  not 
the  flanges  is  the  weak  part  of  the  ordinary  G  frame  and  that 
reinforcement  of  this  by  increasing  its  thickness  or  by  the  addition 
of  radial  ribs  is  the  rational  method  of  treatment. 

It  is  also  evident  that  the  experiments  just  quoted  substanti- 
ate many  of  the  conclusions  reached  by  Professor  Jenkins. 

The  application  of  the  common  formulas  for  beams  to  the 
results  given  in  Table  XIV  gave  values  for  the  unit  stress  which 
are  contradictory  and  misleading.  The  more  complicated  formula 
of  Bach1  was  equally  unsatisfactory. 

Further  experiments  may  lead  to  empirical  formulas,  which  will 
answer  for  all  ordinary  purposes  of  design. 

20.  Principles  of  Design.  —  In  designing  a  frame  for  a  punch  or 
shear  press  similar  to  those  shown  in  Figs.  7  and  9,  attention  must 
be  paid  to  the  stiffness  as  well  as  the  strength,  since  any  sensible 
deflection  or  distortion  will  cause  trouble  with  the  dies  and 
punches  which  do  the  cutting.  In  future  experiments,  it  is  de- 
sirable that  careful  attention  be  paid  to  the  relative  stiffness  of 
various  sections.  It  is  probable  that  the  thickness  of  the  web 
and  the  weight  of  the  outside  flange  have  much  to  do  with  stiff 
ness  and  that  these  have  sometimes  been  neglected  when  strength 
alone  has  been  considered. 

Formula  (13)  may  be  put  in  the  following  shape  for  convenient 
use: 

w= 


ly    1  (14) 

I  +  A 

where 

W  =  pressure  between  dies. 
S  —safe  tensile  stress  on  material. 
I    =  perpendicular  distance  from  line   of  pressure  to 
neutral  axis  of  section. 

=  section  modulus  (tension  side). 

y 

1  Bach's  Elasticital  and  Festigheit. 


48  MACHINE  DESIGN 

A  =area  of  section. 

The  formula  applies  to  any  horizontal  section  as  AB,  Fig.  10. 
For  any  inclined  section,  the  equation  becomes: 


w 


ly    cos  a  (14a) 

7~*~A~ 
where 

a  =  angle   made   by  the    section   with   the   horizontal. 

For  any  section  parallel  to  the  line  of  pressure,  the  second 
term  in  the  denominator  disappears  and  the  formula  is  the  same 
as  for  an  ordinary  cantilever  beam. 

The  stress  in  the  web  at  any  section  in  the  curved  spine  of  the 
frame  is  largely  tension  and  may  account  in  part  for  a  fracture 
like  No.  5  in  Fig.  10. 

This  may  be  illustrated  in  Fig.  10  by  considering  the  outer  and 
inner  flanges  as  separate  members  connected  by  radial  lattice 
work.  It  is  evident  that  pressure  tending  to  open  the  gap 
would  also  tend  to  move  the  flanges  further  apart,  increase  the 
distance  A  B  and  subject  the  radial  lattice  bars  to  tension.  To 
meet  this  condition,  some  manufacturers  introduce  radial  ribs  as 
shown  in  Fig.  7.1  Some  manufacturers  provide  means  for  rein- 
forcing the  gap  in  a  shear  or  punch  frame  by  steel  stays  which  can 
be  attached  when  especially  heavy  work  is  to  be  done..  Fig.  13 
illustrates  a  frame  of  this  character.  The  machine  has  a  60-in. 
gap  and  is  capable  of  punching  a  2^-in.  hole  in  1^-in.  iron  or 
shearing  a  bar  of  flat  iron  1^  X8  in. 

There  is  always  present  the  possibility  that  the  neutral  axis  of 
any  section  does  not  exactly  coincide  with  its  center  of  gravity, 
especially  in  the  curved  portion,  but  the  uncertainties  of  the 
material  itself  outweigh  any  consideration  of  this  sort. 

Straight  Frames.  —  Frames  which  have  a  straight  spine  like 
those  of  drill  presses,  slotting  machines  and  profiling  machines, 
are  similar  in  condition  to  type  (c),  Fig.  9,  and  have  a  uniform 
bending  moment  in  the  straight  part  combined  with  uniform 
tension.  The  condition  is  that  of  a  column  in  tension  with 
eccentric  loading  and  the  deflection  is  usually  the  thing  to  be 
considered  rather  than  the  strength.  This  may  be  illustrated  by 
the  ordinary  iron  clamp  such  as  is  used  in  foundries  and 

1  For  further  discussion  of  this  point,  see  Professor  Jenkin's  paper. 


DESIGN  OF  FRAMES  49 

pattern  shops  and  which  sometimes  assumes  the  shape  shown 
in  Fig.  14.  Practically  the  frame  is  more  likely  to  break  at  the 
curved  portion  joining  the  column  or  spine  with  the  horizontal 
members.  This  is  doubtless  due  to  the  shrinkage  strains  caused 
by  the  profile  at  this  point. 


FIG.  14. 

The  frame  of  a  side-crank  engine  is  a  good  example  of  the 
straight  frame  with  eccentric  loading.  The  points  of  rupture  are 
apt  to  be  at  the  junction  of  the  frame  with  the  cylinder  flange  or 
near  the  main  bearing. 

REFERENCES 

Modern  American  Machine  Tools.     Benjamin. 

Evolution  of  Machine  Tools.     Cass.,  Dec.,  1898;  Cass.,  Sept.,  1904. 

Things  that  are  Usually  Wrong.     Am,  Mach.,  Mar.  9,  1905. 

Design  of  Boring  Mill.     Am.  Mach.,  Mar.  8,  1906. 

Cast  Iron  in  Machine  Frames.     Am.  Mach.,  Oct.  24,  1907. 

Design  of  Machine  Frames.     Mchy.,  Aug.,  1908. 


CHAPTER  III 


CYLINDERS  AND  PIPES 

21.  Thin  Shells. — Let  Fig.  15  represent  a  section  of  a  thin  shell, 
like  a  boiler  shell,  exposed  to  an  internal  pressure  of  p  pounds  per 
square  inch.  Then,  if  we  consider  any  diameter  A B,  the  total 
upward  pressure  on  the  upper  half  of  the  shell  will  balance  the 
total  downward  pressure  on  the  lower  half  and  tend  to  sepa- 
rate the  shell  at  A  and  B  by  tension. 


FIG.  15. 


Let 


d=  diameter  of  shell  in  inches 
r=  radius  of  shell  in  inches 
I  =  length  of  shell  in  inches 
t=  thickness  of  shell  in  inches 
S=  tensile  strength  of  material. 

Draw  the  radial  line  CP  to  represent  the  pressure  on  the  element 
P  of  the  surface. 

Area  of  element  at  P  =  IrdO. 
Total  pressure  on  element  —  plrdd. 
Vertical  pressure  on  element  =  plr  sin  ddd. 

Total  vertical  pressure  on  APB  =      \   plr  sin  Odd  =2plr. 

7Tc/ 

50 


THIN  SHELLS  51 

The  area  to  resist  tension  at  A  and  B  =  2tl  and  its  total  strength 


Equating  the  pressure  and  the  resistance 

2tlS  =  2plr 

pr_pd 
-~S~2S 

The  total  pressure  on  the  end  of  a  closed  cylindrical  shell  = 
nr2p  and  the  resistance  of  the  circular  ring  of  metal  which  resists 
this  pressure  =  2nrtS. 
Equating: 


Therefore  a  shell  is  twice  as  strong  in  this  direction  as  in  the 
other.  Notice  that  this  same  formula  would  apply  to  spherical 
shells. 

In  calculating  the  pressure  due  to  a  head  of  water  equal  h, 
the  following  formula  is  useful: 

p  =  0.434/z,  (17) 

In  this  formula  h  is  in  feet  and  p  in  pounds  per  square  inch. 

PROBLEMS 

1.  A  cast-iron  water  pipe  is  10  in.  in  internal  diameter  and  the  metal  is 
f  in.  thick.     What  would  be  the  factor  of  safety,  with  an  internal  pressure 
due  to  a  head  of  water  of  250  ft.? 

2.  What  would  be  the  stress  caused  by  bending  due  to  weight,  if  the  pipe 
in  Ex.  1  were  full  of  water  and  24  ft.  long,  the  ends  being  merely  supported? 

3.  A  standard  lap-welded  steam  pipe,  6  in.  in  nominal  diameter  is  0.28 
in.  thick  and  is  tested  with  an  internal  pressure  of  500  Ib.  per  square  inch. 
What  is  the  bursting  pressure  and  what  is  the  factor  of  safety  above  the  test 
pressure,  assuming  $  =  40,000? 

22.  Thick  Shells.  —  There  are  several  formulas  for  thick  cylin- 
ders and  no  one  of  them  is  entirely  satisfactory.  It  is,  however, 
generally  admitted  that  the  tensile  stress  caused  by  internal 
pressure  in  such  a  cylinder  is  greatest  at  the  inner  circumference 
and  diminishes  according  to  some  law  from  there  to  the  exterior 
of  the  shell.  This  law  of  variation  is  expressed  differently  in  the 
different  formulas. 

Barlow's  Formulas.  —  Here  the  cylinder  diameters  are  assumed 


52  MACHINE  DESIGN 

to  increase  under  the  pressure,  but  in  such  a  way  that  the  volume 
of  metal  remains  constant.     Experiment  has  proved  that  in 
extreme   cases  this  last   assumption  is  incorrect.     Within  the 
limits  of  ordinary  practice  it  is,  however,  approximately  true. 
Let  dy  and  d2  be  the  interior  and  exterior  diameters  in  inches 

and  let  t=   2       1  be  the  thickness  of  metal. 

2i 

Let  I  be  the  length  of  cylinder  in  inches. 

Let  Sl  and  $2  be  the  tensile  stresses  in  pounds  per  square  inch 
at  inner  and  outer  circumferences. 

The  volume  of  the  ring  of  metal  before  the  pressure  is  applied 
will  be: 


and  if  the  two  diameters  are  assumed  to  increase  the  amounts  x1 
and  x2  under  pressure  the  final  volume  will  be  : 


Assuming  the  volume  to  remain  the  same: 

d2*-d*=(d2+x2y-(dl+xl)* 

Neglecting  the  squares  of  x±  and  x2  this  reduces  to 

dlxi  =  d2x2 

or  the  distortions  are  inversely  as  the  diameters. 
The  unit  deformations  will  be  proportional  to 


and  the  stresses  Sl  and  S2  will  be  in  the  same  ratio: 

S±  =  x1d2  =  dl 
S2     x2d,     d? 

or  the  stresses  vary  inversely  as  the  squares  of  the  diameters. 
Let  S  be  the  stress  at  any  diameter  d,  then: 

S.d*     S.r* 
S=    \2  2  (where  r  is  radius) 

and  the  total  stress  on  an  element  of  the  area  l.dr  is: 


THICK  SHELLS  53 

7  «7 

Integrating  this  expression  between  the  limits  ~  and--  for  r  and 

—  2i 

multiplying  by  2  we  have: 


Equating  this  to  the  pressure  which  tends  to  produce  rupture, 
pdl,  where  p  is  the  internal  unit  pressure,  there  results: 


2  Sf/ 

The  formula  (15)  for  thin  shells  gives  p  =  —-=-• 

a 

By  comparing  this  with  formula  (18)  it  will  be  seen  that  in 
designing  thick  shells  the  external  diameter  determines  the  work- 
ing pressure  or: 


Lame's   Formula.  —  In   this    discussion   each    particle    of   the 
metal  is  supposed  to  be  subjected  to  radial  compression  and  to 


FIG.  16. 

tangential  and  longitudinal  tension  and  to  be  in  equilibrium 
under  these  stresses. 

Using  the  same  notation  as  in  previous  formula: 

d/4-d* 
^~-'Pt 


for  the  maximum  stress  at  the  interior,  and 


54  MACHINE  DESIGN 

s*=TF=ir*Pl  (20> 

for  the  stress  at  the  outer  surface. 

Fig.  16  illustrates  the  variation  in  S  from  inner  to  outer 
surface. 

Solving  for  d2  in  (19)  we  have 

^S/fzfj-  (21) 

A  discussion  of  Lamp's  formula  may  be  found  in  most  works  on 
strength  of  materials. 

PROBLEMS 

1.  A  hydraulic  cylinder  has  an  inner  diameter  of  12  in.,  a  thickness  of  4 
in.  and  an  internal  pressure  of  1500  Ib.  per  square  inch.     Determine  the 
maximum  stress  on  the  metal  by  Barlow's  and  Lame's  formulas. 

2.  Design  a  cast-iron  cylinder  8  in.  internal  diameter  to  carry  a  working 
pressure  of  1200  Ib.  per  square  inch  with  a  factor  of  safety  of  10. 

3.  A  cast-iron  water  pipe  is  1  in.  thick  and  18  in.  internal  diameter. 
Required  head  of  water  which  it  will  carry  with  a  factor  of  safety  of  6. 

23.  Steel  and  Wrought-iron  Pipe. — Pipe  for  the  transmission 
of  steam,  gas  or  water  may  be  made  of  wrought  iron  or  steel. 
Cast  iron  is  used  for  water  mains  to  a  certain  extent,  but  its  use 
for  either  steam  or  gas  has  been  mostly  abandoned.  The  weight 
of  cast-iron  pipe  and  its  unreliability  forbid  its  use  for  high 
pressure  work. 

Wrought-iron  pipe  up  to  and  including  1  in.  in  diameter  is 
usually  butt-welded,  and  above  that  is  lap-welded.  Steel  pipes 
may  be  either  welded  or  may  be  drawn  without  any  seam. 
Electric  welding  has  been  successfully  applied  to  all  kinds  of  steel 
tubing,  both  for  transmitting  fluids  and  for  boiler  tubes. 

The  tables  on  pp.  56  to  61  are  taken  by  permission  from  the 
catalogue  of  the  Crane  Company  and  show  the  standard  dimen- 
sions for  steam  pipe  and  for  boiler  tubes. 

Ordinary  standard  pipe  is  used  for  pressures  not  exceeding 
100  Ib.  per  square  inch,  extra  strong  pipe  for  the  pressures  pre- 
vailing in  steam  plants  where  compound  and  triple  expansion 
engines  are  used,  while  the  double  extra  is  employed  in  hydrau- 
lic work  under  the  heavy  pressures  peculiar  to  that  sort  of 
transmission. 


BOILER  TUBES  55 

Tests  made  by  the  Crane  Company  on  ordinary  commercial 
pipe  such  as  is  listed  in  Table  XV  showed  the  following  pressures: 

8  in.  diam  ........   2,000  Ib.  per  square  inch. 

10  in.  diam  ........    2,300  Ib.  per  square  inch. 

12  in.  diam  ........    1,500  Ib.  per  square  inch. 

The  pipe  was  not  ruptured  at  these  pressures. 

24.  Strength  of  Boiler  Tubes.  —  When  tubes  are  used  in  a  so- 
called  fire-tube  boiler  with  the  gas  inside  and  the  water  outside, 
they  are  exposed  to  a  collapsing  pressure. 

The  same  is  true  of  the  furnace  flues  of  internally  fired  boilers. 
Such  a  member  is  in  unstable  equilibrium  and  it  is  difficult  to 
predict  just  when  failure  will  occur. 

Experiments  on  small  wrought-iron  tubes  have  shown  the 
collapsing  pressure  to  be  about  80  per  cent  of  the  bursting- 
pressure.  With  short  tubes  set  in  tube  sheets  the  length  would 
have  considerable  influence  on  the  strength,  but  ordinary  boiler 
tubes  collapsing  at  the  middle  of  the  length  would  not  be  in- 
fluenced by  the  setting. 

The  strength  of  such  tubes  is  proportional  to  some  function  of 

-,  where  t  is  the  thickness  and  d  is  the  diameter.  The  formulas 
a 

heretofore  in  use  are  very  limited  in  their  application,  being 
founded  on  experiments  covering  but  a  few  diameters  and 
thicknesses. 

Fairbairn's  formula  is  the  oldest  and  best  known  of  these  and 
was  established  by  him  as  a  result  of  experiments  on  wrought- 
iron  flues  not  over  5  ft.  in  length  and  having  relatively  thin 
walls. 

p  =  9,672,000         - 


all  dimensions  being  in  inches  and  p  being  the  collapsing  pressure. 
D.  K.  Clark  gives  for  large  iron  flues  the  following  formula: 

200,000*' 
~~d^~ 

where  P  is  the  collapsing  pressure  in  pounds  per  square  inch. 
These  flues  had  diameters  varying  from  30  in.  to  50  in.  and  thick- 
ness of  metal  from  f  in.  to  y7^-  in. 


56 


MACHINE  DESIGN 


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58 


MACHINE  DESIGN 


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HYDRAULIC  PIPE 


59 


TABLE  XVII 

WROUGHT-IRON  AND  STEEL  DOUBLE  EXTRA  STRONG  PIPE 
Table  of  Standard  Dimensions 

Ill 
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MACHINE  DESIGN 


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BOILER  TUBES 


61 


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62  MACHINE  DESIGN 

In  1906,  Professor  R.  T.  Stewart  reported  to  the  American 
Society  of  Mechanical  Engineers  some  very  comprehensive 
and  interesting  experiments  on  lap-welded  boiler  tubes  of 
Bessemer  steel.1 

The  tests  were  conducted  at  the  works  of  the  National  Tube 
Company  on  tubes  manufactured  by  that  firm  and  were  in 
progress  for  four  years. 

Two  series  of  experiments  were  made — one  on  tubes  8f  in. 
outside  diameter  of  different  thicknesses  and  of  different  lengths, 
for  the  purpose  of  testing  the  applicability  of  existing  formulas 
to  tubes  of  this  character;  one  on  tubes  20  ft.  long  and  of  different 
diameters  and  thicknesses  for  the  purpose  of  establishing  empirical 
formulas  for  the  strength  of  such  tubes. 

The  formulas  of  Fairbairn,  Clark,  Unwin,  Grashof,  etc.,  were 
tested  by  comparison  with  the  results  of  the  first  series  of  experi- 
ments and  were  all  found  inapplicable,  sometimes  giving  less 
than  one-third  the  actual  collapsing  pressure. 

The  general  conclusions  reached  by  Professor  Stewart  are  thus 
stated  by  him: 

"1.  The  length  of  tube,  between  transverse  joints  tending  to 
hold  it  to  a  circular  form,  has  no  practical  influence  upon  the 
collapsing  pressure  of  a  commercial  lap-welded  steel  tube,  so 
long  as  this  length  is  not  less  than  about  six  diameters  of  tube. 

2.  The  formulas,  as  based  upon  the  present  research,  for  the 
collapsing  pressure  of  modern  lap-welded  Bessemer  steel  tubes, 
are  as  follows: 


P  =  1000(1  -  ^l  -  1600     -)  (A) 

P  =  86,670-^-1386.  (B) 

a 

Where  P  =  collapsing  pressure,  pounds  per  square  inch 

d  =  outside  diameter  of  tube  in  inches 

t  =  thickness  of  wall  in  inches. 
Formula  (A)  is  for  values  of  P  less  than  581  lb.,  or  for  values 

of  j  less  than  0.023,  while  formula  (B)  is  for  values  greater  than 
these. 

1  Trans.  A.  S.  M.  E.,  Vol.  XXVII. 


STEEL  TUBES  63 

These  formulas,  while  strictly  correct  for  tubes  thajt  are  20  ft. 
in  length  between  transverse  joints  tending  to  hold  them  to  a 
circular  form,  are,  at  the  same  time,  substantially  correct  for 
all  lengths  greater  than  about  six  diameters. 

They  have  been  tested  for  seven  diameters,  ranging  from 
3  to  10  in.,  in  all  obtainable  thicknesses  of  wall,  and  are  known  to 
be  correct  for  this  range. 

3.  The  apparent' fiber  stress  under  which  the  different  tubes 
failed  varied  from  about  7000  Ib.  for  the  relatively  thinnest  to 
35,000  Ib.  per  square  inch  for  the  relatively  thickest  walls. 

Since  the  average  yield-point  of  the  material  was  37,000  and 
the  tensile  strength  58,000  Ib.  per  square  inch,  it  would  appear 
that  the  strength  of  a  tube  subjected  to  a  collapsing  fluid  pressure 
is  not  dependent  alone  upon  either  the  elastic  limit  or  ultimate 
strength  of  the  material  constituting  it." 

The  following  tables  are  condensed  from  those  published  by 
Professor  Stewart  and  give  average  dimensions  and  pressures 
for  each  size  tested,  each  result  being  the  average  of  five  tubes: 

The  reader  is  referred  to  the  published  paper  for  further 
details  of  this  most  valuable  contribution  to  a  hitherto  neglected 
subject. 

25.  Theory. — In  January,  1911,  Professor  Stewart  presented  a 
discussion  of  the  theory  of  collapsed  tubes  based  on  the  experi- 
ments above  described.1  Considering  a  ring  or  annulus  of  the 
tube  1  in.  long  near  the  middle  of  its  length,  he  treats  each  half  of 
the  ring  as  a  column  fixed  at  both  ends  and  compressed  uniformly 
along  its  center  line,  abc,  Fig.  17. 

The  ring  is  subjected  to  a  uniform  radial  external  pressure  of 
p  pounds  per  square  inch  and  is  therefore  in  the  same  condition 
as  the  thin  shell  in  Art.  21  except  that  the  resultant  stress  is  now. 
compression  instead  of  tension.  By  equation  (15), 

~     pr_pd 
and 


2tS  ,  . 

(a) 


1  Trans.  A.  S.  M.  E.,  Vol.  XXXIII. 


64 


MACHINE  DESIGN 


TABLE  XIX 

COLLAPSING  PRESSURE  OF  TUBES 


Test 
number 

Average 
outside 
diameter, 
inches 

Average 
thickness  of 
wall, 
inches 

Actual 
length  of 
tube, 
feet 

Collapsing 
pressure, 
pounds  per 
square  inch 

1 

8.643 

0.185 

20.026 

536 

2 

8.653 

0.184 

15.010 

548 

3 

8.656 

0.178 

10  .  002 

548 

4 

8.658 

0.180 

5.006 

592 

5 

8.656 

0.176 

2.512 

977 

6 

8.642 

0.215 

13.140 

847 

7 

8.663 

0.219 

11.801 

835 

8 

8.669 

0.214 

10.007 

845 

9 

8.661 

0.212 

4.997 

907 

10 

8.657 

0.212 

2.507 

1,314 

11 

8.666 

0.267 

19.995 

1,438 

12 

8.652 

0.272 

14.996 

1,540 

13 

8.668 

0.267 

9.993 

1,533 

14 

8.656 

0.268 

4.993 

1,636 

15 

8.662 

0.268 

2.494 

1,784 

16 

8.657 

0.273 

19.387 

1,347 

17 

8.659 

0.275 

14.995 

1,421 

18 

8.671 

0.271 

10.003 

1,541 

19 

8.672 

0.280 

4.997 

1,731 

20 

8.653 

0.269 

2.505 

1,961 

21 

8.656 

0.294 

19.999 

1,686 

22 

8.654 

0.308 

14.987 

1,791 

23 

8.649 

0.305 

9.989 

1,810 

24 

8.654 

0.306 

4.993 

2,073 

25 

8.646 

0.311 

2.509 

2,397 

26 

6.017 

0.128 

20.000 

519 

27 

6.017 

0.131 

20.000 

529 

28 

6.022 

0.167 

20.000 

969 

29 

6.026 

0.166 

20.000 

924 

30 

6.032 

0.163 

20.000 

917 

31 

6.033 

0.170 

20.000 

,007 

32 

6.023 

0.189 

20  .  000 

,318 

33 

6.021 

0.212 

20.000 

,457 

34 

6.015 

0.206 

20  .  000 

,555 

35 

6.022 

0.186 

20.000 

,188 

36 

6.032 

0.263 

20.000 

2,139 

STEEL  TUBES 


65 


TABLE  XIX—  (Continued) 
COLLAPSING  PEESSURE  OF  TUBES 


Test 
number 

Average 
outside 
diameter, 
inches 

Average 
thickness 
of  wall, 
inches 

Actual 
length  of 
tube, 
feet 

Collapsing 
pressure, 
pounds  per 
square  inch 

37 

6.034 

0.264 

20.000 

2,381 

38 

6.654 

0.164 

20.000 

678 

39 

6.684 

0.200 

20.000 

1,184 

40 

6.666 

0.253 

20.000 

2,081 

41 

7.044 

0.160 

20  .  000 

563 

42 

7.050 

0.242 

20.000 

1,680 

43 

6.661 

0.154 

20.000 

563 

44 

6.655 

0.269 

20.100 

2,214 

45 

6.681 

0.249 

20.100 

1,745 

46 

6.049 

0.266 

20.110 

2,528 

47 

8.643 

0.185 

20.000 

536 

48 

8.642 

0.215 

14.133 

847 

49 

8.666 

0.267 

19.995 

1,438 

50 

8.657 

0.273 

19.550 

1,347 

51 

8.656 

0.293 

20.000 

1,686 

52 

8.663 

0.305 

20.100 

1,756 

53 

8.673 

0.354 

20.080 

2,028 

54 

6.987 

0.279 

20.170 

2,147 

55 

7.011 

0.160 

20.170 

621 

56 

5.993 

0.271 

20.180 

2,487 

57 

10.041 

0.165 

20.180 

225 

58 

10.026 

0.194 

20.110 

383 

59 

10.001 

0.316 

20.180 

1,319 

60 

3.993 

0.119 

20.170 

964 

61 

4.014 

0.175 

20.190 

2,280 

62 

4.026 

0.212 

20.190 

3,170 

63 

4.014 

0.327 

20.100 

5,560 

64 

3.000 

0.109 

20.000 

1,733 

65 

2.994 

0.113 

20.000 

1,962 

66 

2.992 

0.143 

20.000 

2,963 

67 

2.995 

0.188 

20.100 

4,095 

68 

10.779 

0.512 

19.470 

2,585 

69 

12.790 

0.511 

19.960 

2,196 

70 

13.036 

0.244 

20.000 

463 

66 


MACHINE  DESIGN 


This  stress  is  uniform  from  end  to  end  as  is  the  case  with  the 
loaded  straight  column  in  Fig.  18.  Furthermore,  the  character- 
istic shape  assumed  by  the  collapsed  tube,  as  shown  in  dotted 
lines  in  Fig.  17,  has  its  tangents  at  a'  and  c'  parallel  to  their 
original  position  at  a  and  c,  corresponding  to  the  conditions  for 
buckling  of  a  column  with  fixed  ends  shown  by  dotted  lines  in 
Fig.  18. 


V 

\ 
\ 

\       \ 

\       \ 

\       \ 

\ 

\ 

I 


/      I 
I      I 
I 


Let  /  =  length  of  equivalent  column 

r  =  radius  of  gyration  of  section  of  column 

Then  will  l  =  ^(d-t) 

(where  d   =  outer  diameter  of  tube) 
and 


IV     _J_ 
\12     3.464 


By  Professor  Stewart's  formula  (B) 

P  =  86,670^-1386 

From   (a)   and   (B)   by  equating: 

—  =86,670^-1386 


(b) 


(c) 


STEEL  TUBES  67 

and 

£=43,335-693^  (d) 

From  (b)  : 

1--I+1 

t         Tit 

Substituting  value  of  t  from   (c)  : 

d.  =  ~?L     +1=0.1838   ~  +  l  (e) 

t     3.4647ZT  r 

Substituting  this  value  of  -  in   (d)   and  reducing: 

L 

S  =  42,642-  127.4  -  (24) 

corresponding   to  the    straight   line  formula  for    columns    (see 
Table  la). 

Professor  Stewart  suggests  as  a  substitute  for  formula  (A) 
p.  62,  the  following: 


P  =  50,210,000  (G) 

26.  Tube  Joints.  —  The  failure  of  boiler  tubes,  especially  of  those 
having  water  or  steam  pressure  inside,  is  frequently  due  to 
slipping  of  the  tube  in  the  plate  or  fitting  to  which  it  joins.  Such 
tubes  are  expanded  in  the  plate  by  the  use  of  a  roller  or  Dudgeon 
expander  and  are  sometimes  flared  or  beaded  on  the  outside  for 
additional  security.  Under  pressure,  the  tubes  often  slip  in  the 
holes  so  as  to  cause  failure  of  the  joint  or  at  least  leakage  of  the 
contained  fluid. 

Some  experiments  made  by  Professors  O.  P.  Hood  and  G.  L. 
Christiansen  were  reported  by  them  in  1908  and  give  the  most 
reliable  information  on  this  subject.1 

The  tests  were  made  on  3-in.,  twelve-gage,  cold  drawn  Shelby 
tubes  rolled  into  holes  in  plates  of  various  thicknesses  and  reamed 
in  various  shapes.  Some  of  the  tubes  were  flared  outside  the 
plate  and  some  not. 

Initial  slip  occurred  at  total  pressures  of  from  5000  to  10,000 
Ib.  or  from  one-sixth  to  one-third  the  elastic  limit  of  the  material 

1  Trans.  A.  S.  M.  E.,  Vol.  XXX. 


68  MACHINE  DESIGN 

of  the  tube.  The  ultimate  holding  power  was  usually  about 
double  the  slipping  load. 

The  coefficient  of  friction  varied  from  26  to  35  per  cent,  assum- 
ing the  elastic  limit  to  vary  between  30,000  and  40,000  Ib. 

The  total  friction  per  square  inch  of  bearing  area  was  about 
750  Ib.  Various  degrees  of  rolling  and  various  forms  of  tapered 
hole  did  not  seem  to  affect  the  initial  slipping  load  materially. 
Serrating  the  bearing  surface  of  the  hole  had  a  very  marked 
effect,  raising  the  initial  slipping  load  in  some  instances  as  high 
as  40,000  to  45,000  Ib.,  or  more  than  the  elastic  limit  of  the  tube. 

The  slipping  point  of  the  tube  bears  a  certain  analogy  to  the 
yield-point  in  metals  and  the  diagrams  of  pressure  and  slip  much 
resemble  the  stress-strain  diagrams  of  soft  steel. 

It  is  apparent  from  these  experiments  that  overrolling  has  no 
advantages  and  that  flaring  the  tubes  will  not  prevent  leakage. 

The  fact  that  ordinarily  slipping  will  occur  at  a  pressure  well 
inside  the  elastic  limit  of  the  material  shows  that  timely  warning 
will  be  given  by  leakage  before  there  is  any  danger  of  failure. 

27.  Tubes  under  Concentrated  Loads. — In  1893,  the  author 
made  some  experiments  on  steel  hoops  to  determine  the  strength 
and  stiffness  under  a  concentrated  load  applied  in  the  direction 
of  a  diameter.1 

Large  steel  tubes  with  relatively  thin  walls  are  sometimes 
exposed  to  external  compression  at  the  point  of  support  causing 
distortion  and  occasionally  permanent  injury. 

The  hoops  tested  were  made  of  mild  steel  boiler  plate,  having 
a  tensile  strength  of  60,000  Ib.  and  a  modulus  of  elasticity  of 
30,000,000,  cut  into  strips  2.5  in.  wide,  bent  to  a  circular  form 
and  welded.  Each  hoop  was  compressed  laterally  in  a  testing 
machine  until  failure  occurred,  vertical  and  horizontal  diameters 
being  measured  at  regular  intervals. 

Regarding  the  hoop  as  composed  of  two  semi-circular  columns 
fixed  at  the  ends  and  each  having  a  constant  deflection  of  one- 
half  the  mean  diameter,  it  is  evident  that  a  treatment  is  allowable 
similar  to  that  used  in  Rankine's  formula  for  columns  (for- 
mula (12)). 

The  increase  in  deflection  for  loads  inside  the  elastic  limit  is 
small  compared  with  the  length  of  the  hoop  radius. 

lJour.  Assoc.  Eng.  Soc.,  Dec.,  1893. 


STEEL  HOOPS  69 

Let  P  =  load  in  pounds  at  elastic  limit  • 

D  =  inner  horizontal  diameter  in  inches 
6  =  breadth  of  hoop  in  inches 
t  =  thickness  of  ring 
S  —  stress   on   inner   fibers   at   extremity   of   horizontal 

diameter. 
Then  as  in  Rankine's  formula: 


Where  M  is  the  bending  moment  at  extremity  of  horizontal 
diameter. 

Assume  M  =  kPD. 
Then 


p 


where  q  =  empirical  constant. 

The  average  value  of  q  as  determined  by  experiment  was 

q  =  0.946. 
Substituting  this  value  in  (b)  and  solving  for  P}  we  have: 

2btS 

"1  +  0.946.D' 
t 

Table  XX  gives  the  principal  data  and  results  of  experiment. 

In  determining  the  value  of  q  from  the  experiments,  S  was 
assumed  to  be  the  same  as  the  elastic  limit  in  compression  of  a 
straight  specimen  of  the  same  metal. 

The  limited  number  of  hoops  tested  and  the  method  of  their 
construction  forbids  the  application  of  formula  (25)  to  general 
cases  of  this  character.  It  is  offered  here  merely  as  a  guide  in 
design. 

28.  Pipe  Fittings. — Steam  pipe  up  to  and  including  pipe  2  in. 
in  diameter  is  usually  equipped  with  screwed  fittings,  including 
ells,  tees,  couplings,  valves,  etc. 

Pipe  of  a  larger  size,  if  used  for  high  pressures,  should  be  put 
together  with  flanged  fittings  and  bolts.  One  great  advantage  of 


70 


MACHINE  DESIGN 


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PIPE  FITTINGS  71 

the  latter  system  is  the  fact  that  a  section  of  pipe  can' easily  be 
removed  for  repairs  or  alterations. 

Small  connections  are  usually  made  of  cast  iron  or  malleable 
iron.  While  the  latter  are  neater  in  appearance  they  are  more 
apt  to  stretch  and  cause  leaky  joints.  The  larger  fittings  are 
made  of  cast  iron  or  cast  steel.  Such  fittings  can  be  obtained  in 
various  weights  and  thicknesses,  to  correspond  to  those  grades  of 
pipe  listed  in  the  tables. 

The  designer  should  have  at  hand  catalogues  of  pipe  fittings 
from  the  various  manufacturers,  as  these  will  give  in  detail  the 
proportions  of  all  the  different  connections. 

For  pressures  not  exceeding  100  Ib.  per  square  inch  rubber  and 
asbestos  gaskets  can  be  used  between  the  flanges,  but  for  higher 
pressures  or  for  superheated  steam,  corrugated  metallic  gaskets 
are  necessary. 

In  1905  some  very  interesting  experiments  on  the  strength  of 
standard  screwed  elbows  and  tees  were  made  by  Mr.  S.  M. 
Chandler,  a  graduate  of  the  Case  School,  and  published  by  him 
in  Power  for  October,  1905. 

The  fittings  were  taken  at  random  from  the  stock  of  the  Pitts- 
burg  Valve  and  Fittings  Co.,  and  three  of  each  size  were  tested 
to  destruction  by  hydraulic  pressure. 

The  following  table  gives  a  summary  of  the  results  obtained. 
The  values  which  are  starred  in  the  table  were  obtained  from 
fittings  which  had  purposely  been  cast  with  the  core  out  of  center 
so  as  to  make  one  wall  thinner  than  the  other.  These  values  are 
not  included  in  the  averages. 

These  tests  show  a  large  apparent  factor  of  safety  for  any 
pressures  to  which  screwed  fittings  are  usually  subjected. 

The  failure  of  such  fittings  in  practice  must  be  attributed  to 
faulty  workmanship  in  erection,  such  as  screwing  too  tight,  lack 
of  allowance  for  expansion  and  poor  drainage. 

The  average  tensile  strength  of  the  cast  iron  used  in  the  above 
fittings  was  20,000  Ib.  per  square  inch. 

29.  Flanged  Fittings. — In  1907,  the  Crane  Company  published 
the  results  of  a  series  of  tests  made  on  flanged  tees  and  ells 
manufactured  by  that  company.1 

The  fittings  were  tested  by  hydraulic  pressure,  a  blank  flange 

1  Valve  World,  Nov.,  1907. 


72 


MACHINE  DESIGN 


TABLE  XXI 

BURSTING   STRENGTH   OF  STANDARD   SCREWED  FITTINGS,   PRESSURES   IN 
POUNDS  PER  SQUARE  INCH 


Size 

Elbows 

Average 

2* 

3,500 

3,300 

3,400 

3,400 

3 

2,400 

2,600 

2,100* 

2,500 

3* 

2,100 

1,700* 

2,400 

2,250 

4 

2,800 

2,500 

2,500 

2,600 

4* 

2,000* 

2,600 

2,600 

2,600 

5 

2,600 

2,500 

2,500 

2,533 

6 

2,600 

2,200 

2,300 

2,367 

'    7 

1,800 

2,100 

1,900* 

1,950 

8 

1,700 

1,600 

1,700 

1,667 

9 

1,800 

1,800 

1,900 

1,833 

10 

1,800 

1,700 

1,600 

1,700 

12 

1,100 

1,200 

900* 

1,150 

Size 

Tees 

Average 

li 

3,400 

3,300 

3,300 

3,333 

u 

3,400 

3,200 

2,800* 

3,300 

2 

2,500 

2,800 

2,500 

2,600 

2i 

2,400 

2,100* 

2,500 

2,450 

3 

1,400* 

1,900 

,800 

,850 

3i 

1,200* 

1,500 

,800 

,650 

4 

,800 

2,100 

,700 

,867 

4* 

,100* 

1,400 

,400 

,400 

5 

,700 

1,300* 

,500 

,600 

6 

,400 

1,500 

,100* 

,450 

7 

,400 

1,400 

,500 

,433 

8 

,200* 

1,400 

,300 

,350 

9 

,300 

1,400 

1,200 

,300 

10 

1,100 

1,300 

1,200 

1,200 

12 

1,100 

1,000 

1,100 

1,067 

*  Made  with  eccentric  core. 


PIPE  FITTINGS 


73 


74 


MACHINE  DESIGN 


being  used  to  close  the   opening.     Two   materials   were   tried, 
cast  iron  having  an  average  tensile  strength  of  22;000  Ib.  per 
square  inch  and  ferro  steel  having  a  strength  50  per  cent  greater. 
The  results  are  given  as  follows: 


TABLE  XXII 

STRENGTH  OF  FLANGED  FITTINGS 

EXTRA  HEAVY  FITTINGS— TEES 


Size 
inches 

Body 
metal 
inches 

Burst  ferro- 
steel  Ib. 
per 
sq.  in. 

Average 

Burst     cast 
iron  Ib. 
per 
sq.  in. 

Average 

6 

| 

2  700 

6 
6 

8 

I 

1 

13 

2,500 
3,000 
2  100 

2,733 

1,675 
1,700 

1,687 

8 

1  6 

44 

2  250 

8 

13 

2  250 

8 

13 

2  100 

8 

1  6 

14 

2  500 

1,200 

8 
10 

H 

15- 

2,300 
2  200 

2,250 

1,500 

1,350 

10 

1  6 
15 

2  200 

1,225 

10 

.15 

2  100 

1  300 

10 

16 

14 

2  000 

1,200 

10 
12 

*& 
I 

2,300 
2  200 

2,160 

1,500 

1,306 

12 

I 

2  100 

1,100 

12 

1 

2  000 

1  400 

12 

I 

2  000 

1  500 

12 

i 

2  100 

1  450 

12 
14 

1 

1,800 
1  900 

2,033 

1,450 

1,380 

14 
16 
16 

18 

H 

1T36 

IA 
11 

1,750 
1,700 
1,700 
1  600 

1,825 
1,700 

1,100 
1,050 
1,000 

1,100 
1,025 

18 
20 

H 

1    5 

1,300 
1  400 

1,450 

600 

600 

20 
24 

J-16 
1* 
1* 

1,150 
1,300 

1,275 
1,300 

750 
700 

750 
700 

PIPE  FITTINGS 


75 


TABLE    XXII— (Continued) 
EXTRA  HEAVY  FITTINGS— ELLS 


Size 
inches 

Body 
metal 
inches 

Burst  ferro- 
steel     Ib. 
per 
sq.  in.  . 

Average 

Burst  cast 
iron  Ib. 
per 
sq.  in. 

Average 

6 

| 

2,800 

6 

| 

3,500 

2,350 

6 

8 

i 

if 

3,500 
2,700 

3,266 

2,200 
1,700 

2,275 

8 

44 

2,800 

1,600 

8 

13 

2,800 

1,500 

8 
10 

If 
0 

2,600 
2  550 

2,725 

1,700 
1,625 

1,625 

10 
10 
12 

i! 

H 

i 

2,000 
2,500 
2  000 

2,350 

1,400 
1,600 
1,275 

1,541 

12 
12 
14 

i 

i 
u 

2,200 
2,200 
1,700 

2,133 

*900 
*700 
900 

1,275 

14 
16 

H 

i& 

2,100 

1,250 
1,250 

1,075 
1,250 

*  Defective,  eliminated  from  total. 

STRENGTH  OF  FLANGED  FITTINGS 
STANDARD  CAST-IRON  FITTINGS— TEES 


Size 
inches 

Body  metal 
inches 

Bursting  cast  iron 
Ib.  per  sq.  in. 

Average 

6 

A 

1,700 

6 

8 

A 
& 

1,500 
1  150 

1,600 

10 

| 

1,100 

12 
12 
14 

It 

It 
I 

700 
850 
700 

775 

16 

1 

750 

STANDARD  ( 

^AST-IRON  FITTINGS—  ELL 

3 

6 
8 
10 

A 

f 
| 

2,000 
1,500 
1  200 



12 
14 
16 

it 

i 
1 

900 
900 
850 

76 


MACHINE  DESIGN 


The  Company  recommends  a  rule  which  may  be  thus  stated : 

p-f  (26) 

where 

p  =  bursting  pressure  in  Ib.  per  square  inch 
S  =  tensile  strength  of  metal 
t  =  thickness  of  wall  in  inches 
d  —  inside  diameter  in  inches 

c=a  constant,  0.65  for  sizes  up  to  12  in.  and  0.60  for  sizes 
above  that. 


FIG.  21. 

A  factor  of  safety  of  from  4  to  8  is  recommended. 
The  fractures  were  of  various  shapes   and  locations.     The 
usual  failure  of  the  tees  was  by  splitting  in  the  plane  of  the  axes 
from  one  flange  to  the  next  adjacent,  Fig.  21. 

About  half  of  the  ells  failed  around  a 
circumference  inside  one  flange  (Fig.  22) 
while  six  failed  by  splitting  on  the  in- 
side of  the  bend. 

The    effect    on   cast-iron   fittings   of 
high   temperatures   such   as  may  occur 
with  the  use  of  superheated  steam  is  not 
clearly    understood.      Professor    Hollis 
i   and  others  report  experiments  on  such 
fittings  which  seem  to  show  some  deteri- 
oration from  this  cause.1 
It  is  probable  that  most  of  the  failures  of  pipe  fittings  in  service 
are  due  to  the  excessive  expansion  and  contraction  of  the  pipe 
lines,  incident  to  the  use  of  high  temperatures,  rather  than  to  the 
direct  effect  of  the  temperature  or  pressure. 
1  Trans.  A.  S.  M.  E.,  Vol.  XXXI. 


FIG.   22. 


CYLINDERS  77 

A  uniform  temperature  of  600  to  700°  fahr.  will  not  injure  the 
cast-iron  material,  but  where  the  temperature  varies  considerably, 
it  is  best  to  use  some  other  metal. 

PROBLEMS 

1.  Determine  the  bursting  pressure  of  a  wrought-iron  steam  pipe  6  in. 
nominal  diameter. 

(a)  If  of  standard  dimensions. 

(6)   If  extra  strong. 

(c)   If  double  extra  strong. 

2.  Compare  the  above  with  the  strength  of  standard  screwed  and  standard 
flanged  elbows  and  tees  of  the  same  size. 

3.  Determine  the  probable  collapsing  pressure  of  a  soft  steel  boiler-tube 
of  2  in.  nominal  diameter. 

4.  Ditto,  if  tube  is  6  in.  in  diameter. 

30.  Steam  Cylinders.  —  Cylinders  of  steam  engines  can  hardly 
be  considered  as  coming  under  either  of  the  preceding  heads. 
On  the  one  hand  the  thickness  of  metal  is  not  enough  to  insure 
rigidity  as  in  hydraulic  cylinders,  and  on  the  other  the  nature  of 
the  metal  used,  cast  iron,  is  not  such  as  to  warrant  the  assump- 
tion of  flexibility,  as  in  a  thin  shell.  Most  of  the  formulas  used 
for  this  class  of  cylinder  are  empirical  and  founded  on  modern 
practice. 

Van  Bur  en's  formula1  for  steam  cylinders  is: 

*  =  .0001pd-Kl5\/d  (27) 

A  formula  which  the  writer  has  developed  is  somewhat  similar 
to  Van  Buren's. 

Let  s'  =  tangential  stress  due  to  internal  pressure. 
Then  by  equation  for  thin  shells 


Let  s"  be  an  additional  tensile  stress  due  to  distortion  of  the 
circular  section  at  any  weak  point. 

Then  if  we  regard  one-half  of  the  circular  section  as  a  beam 
fixed  at  A  and  B  (Fig.  23)  and  assume  the  maximum  bending 
moment  as  at  C  some  weak  point,  the  tensile  stress  on  the  outer 

1  See  Whitham's  "Steam  Engine  Design,"  p.  27. 


78 


MACHINE  DESIGN 


fibers  at  C  due  to  the  bending  will  be  proportional  to 
the  laws  of  flexure,  or 

s»  =  cP* 

where  c  is  some  unknown  constant. 

The  total  tensile  stress  at  C  will  then  be 


by 


Solving  for  c 
Solving  for  t 


St2 
c  =     ^ 


t 

ltd 


cpd2      p2d2 
~!T"h IQS2 


(a) 


(28) 


a  form  which  reduces  to  that  of  equation  (15)  when  c  =  0. 

An  examination  of  several  engine  cylinders  of  standard 
manufacture  shows  values  of  c  ranging  from  .03  to  .10,  with  an 
average  value: 

c  =  .06. 

The  formula  proposed  by  Professor  Barr,  in  his  paper  on 
"Current  Practice  in  Engine  Proportions,"1  as  representing  the 
average  practice  among  builders  of  low-speed  engines  is: 

Z  =  .05d  +  .3  in.  (29) 

In  Kent's  Mechanical  Engineer's  Pocket  Book,  the  following 
formula  is  given  as  representing  closely  existing  practice: 

t  =  .0004dp -\-0.3  in.  (30) 

1  Trans.  A.  S.  M.  E.,  Vol.  XVIII,  p.  741. 


CYLINDERS 


79 


This  corresponds  to  Barr's  formula  if  we  take  p  =  \25  Ib.  per 
square  inch. 

Experiments1  made  at  the  Case  School  of  Applied  Science  in 
1896-97  throw  some  light  on  this  subject.  Cast-iron  cylinders 
similar  to  those  used  on  engines  were  tested  to  failure  by  water 
pressure.  The  cylinders  varied  in  diameter  from  6  to  12  in.  and 
in  thickness  from  ^  to  f  in. 

Contrary  to  expectations  most  of  the  cylinders  failed  by  tearing 
around  a  circumference  just  inside  the  flange  (see  Fig.  24). 

Table  XXIII  gives  a  summary  of  the  results. 

TABLE  XXIII 


Formulas  used 

Pres- 

Thick- 

Lin     f 

Strength  of 

No, 

d 

sure 
P 

ness 
t 

failure 

15 

16 

s=Pft 

a 

test  bar 

a 

12.16 

800 

.70 

Circum..  .  . 

6,940 

3,470 

.046 

18,000  Ib. 

d 

12.45 

700 

.56 

Longi  

7,780 

.047 

24,000  Ib. 

e 

9.12 

1,325 

.61 

Circum..  .  . 

9,900 

4,950 

.048 

24,000  Ib. 

f 

6.12 

2,500 

.65 

Circum..  .  . 

11,800 

5,900 

.055 

24,000  Ib. 

1 

9.58 

600 

.402 

Longi  

7,150 

.049 

24,000  Ib. 

2 

9.375 

1,050 

.573 

Circum..  .  . 

8,590 

4,300 

.055 

24,000  Ib. 

3             9.13 

975 

.596 

Circum..  .  . 

7,470 

3,740 

.072 

24,000  Ib. 

4           12.53 

700 

.571 

Longi  

7,680 

.048 

24,000  Ib. 

5           12.56 

875 

.531 

Circum..  .  . 

10,350 

5,180 

.028 

24,000  Ib. 

Average  of  c  =  .05 

Out  of  nine  cylinders  so  tested,  only  three  failed  by  splitting 
longitudinally. 

This  appears  to  be  due  to  two  causes.  In  the  first  place,  the 
flanges  caused  a  bending  moment  at  the  junction  with  the  shell 
due  to  the  pull  of  the  bolts.  In  the  second  place,  the  fact  that 
the  flanges  were  thicker  than  the  shell  caused  a  zone  of  weakness 
near  the  flange  due  to  shrinkage  in  cooling,  and  the  presence  of 
what  founders  call  ua  hot  spot." 

The  stresses  figured  from  formula  (16)  in  the  cases  where  the 
failure  was  on  a  circumference,  are  from  one-fifth  to  one-sixth 
the  tensile  strength  of  the  test  bar. 

1  Trans.  A.  S.  M.  E.,  Vol.  XIX. 


80 


MACHINE  DESIGN 


FIG.  24. — FRACTURED  CYLINDER. 


FIG.  25. — FRACTURED  CYLINDER. 


CYLINDERS  81 

The  strength  of  a  chain  is  the  strength  of  the  weakest  link,  and 
when  the  tensile  stress  exceeded  the  strength  of  the  metal  near 
some  blow  hole  or  "hot  spot,"  tearing  began  there  and  gradually 
extended  around  the  circumference. 

Values  of  c  as  given  by  equation  (a)  have  been  calculated  for 
each  cylinder,  and  agree  fairly  well,  the  average  value  being 
c  =  .05. 

To  the  criticism  that  most  of  the  cylinders  did  not  fail  by 
splitting,  and  that  therefore  formula^  (a)  and  (22)  are  not  appli- 
cable, the  answer  would  be  that  the  chances  of  failure  in  the  two 
directions  seem  about  equal,  and  consequently  we  may  regard 
each  cylinder  as  about  to  fail  by  splitting  under  the  final  pressure. 

If  we  substitute  the  average  value  of  c  =  .05  and  a  safe  value  of 
£  =  2000,  formula  (28)  reduces  to: 


P2_ 

8000     200  \P      1600' 

An  application  of  the  Crane  formula  for  cast-iron  pipe  fittings 
to  some  of  the  results  in  Table  XXIII  shows  that  the  conditions 
are  similar. 

R  o/ 

Using  the  formula  p  =  '—r-  for  cylinders  (d)  and  (4)  in  the  table, 

we  have  approximately: 

.6X24,OOOX.57 

p=     -isj- 

as  against  an  actual  value  of  700. 

In  a  similar  manner,  testing  cylinder  (1)  in  table,  we  have: 

.65  X  24,000  X. 402 
p  =          -fag-          =654 

as  against  600  in  the  table.  It  will  be  noted  that  these  are  the 
cylinders  which  failed  by  splitting. 

Subsequent  experiments1  made  at  the  Case  School  in  1904 
show  the  effect  of  stiffening  the  flanges  by  brackets. 

Four  cylinders  were  tested,  each  being  10  in.  internal 
diameter  by  20  in.  long  and  having  a  thickness  of  about  f  in. 
The  flanges  were  of  the  same  thickness  as  the  shell  and  were 
reenforced  by  sixteen  triangular  brackets  as  shown  in  Fig.  25. 

The  fractures  were  all  longitudinal  there  being  but  little  of  the 

1  Mchy.,  N.  Y.,  Nov.,  1905. 


82 


MACHINE  DESIGN 


tearing  around  the  shell  which  was  so  marked  a  feature  of  the 
former  experiments.  This  shows  that  the  brackets  served  their 
purpose. 

Table  XXIV  gives  the  results  of  the  tests  and  the  calculated 
values  of  c. 

TABLE  XXIV 

BURSTING  PRESSURE  OF  CAST-IRON  CYLINDERS 


Internal 
diameter 

Average 
thickness 

Bursting 
pressure 

Value 
of  c 

pd 

S=2t 

10.125 

0.766 

1,350 

.0213 

9,040 

10.125 

0.740 

1,400 

.0152 

10,200 

10.125 

0.721 

1,350 

.0126 

9,735 

10.125 

0.720 

1,200 

.0177 

9,080 

Average  value  of  c  =  .0167. 

Comparing  the  values  in  the  above  table  with  those  in  Table 
XXIII  we  find  c  to  be  only  one-third  as  large. 

The  tensile  strength  of  the  metal  in  the  last  four  cylinders,  as 
determined  from  test  bars,  was  only  14,000  Ib.  per  square  inch. 

Comparison  with  the  values  of  S  due  to  direct  tension  as  given 
by  the  formula 

pd 
=  2* 

shows  that  in  a  cylinder  of  this  type  about  one-third  of  the  stress 
is  " accidental"  and  due  to  lack. of  uniformity  in  the  conditions. 
In  Table  XXIII  about  two-thirds  must  be  thus  accounted  for. 

PROBLEMS 

1.  Referring  to  Table  XXIII,  verify  in  at  least  three  experiments  the 
values  of  S  and  c  as  there  given.     Do  the  same  in  Table  XXIV. 

2.  The  steam  cylinder  of  a  Baldwin  locomotive  is  22  in.  in  diameter 
and  1.25  in.  thick.     Assuming  125  Ib.  gage  pressure,  find  the  value  of  c. 
Calculate  thickness  by  Van  Buren's  and  Barr's  formulas. 

3.  Determine  proper  thickness  for  cylinder  of  cast  iron,  if  the  diameter 
is  42  in.  and  the  steam  pressure  120  Ib.  by  formulas  15,  27,  29,  30  and  31. 

4.  The  cylinder  of  a  stationary  engine  has  internal  diameter  =  14  in.  and 
thickness  of  shell  =  1.25  in.     Find  the  value  of  c  for  p  =  120  Ib.  per  square 
inch. 


FLAT  PLATES  83 

31.  Thickness  of  Flat  Plates.  —  An  approximate  formula  for  the 
thickness  of  flat  ^cast-iron  plates  may  be  derived  as  follows: 
Let      1  =  length  of  plate  in  inches 
b  =  breadth  of  plate  in  inches 
t  =  thickness  of  plate  in  inches 
p  =  intensity  of  pressure  in  pounds 
S  =  modulus  of  rupture  pounds  per  square  inch. 
A  plate  which  is  supported  or  fastened  at  all  four  edges  is  con- 
strained so  as  to  bend  in  two  directions  at  right  angles.     Now  if 
we  suppose  the  plate  to  be  represented  by  a  piece  of  basket  work 
with  strips  crossing  each  other  at  right  angles  we  may  consider 
one  set  of  strips  as  resisting  one  species  of  bending  and  the  other 
set  as  resisting  the  other  bending.     We  may  also  consider  each 
set  of  strips  as  carrying  a  fraction  of  the  total  load.     The  equa- 
tion of  condition  is  that  each  pair  of  strips  must  have  a  common 
deflection  at  the  crossing. 

Suppose  the  plate  to  be  divided  lengthwise  into  flat  strips  an 
inch  wide  I  inches  long,  and  suppose  that  a  fraction  p'  of  the  whole 
pressure  causes  the  bending  of  these  strips. 

Regarding  the  strips  as  beams  with  fixed  ends  and  uniformly 
loaded: 

QWl  _p'l2 


_ 

W~ 
and  the  thickness  necessary  to  resist  bending  is: 


In  a  similar  manner,  if  we  suppose  the  plate  to  be  divided  into 
transverse  strips  an  inch  wide  and  b  inches  long,  and  suppose  the 
remainder  of  the  pressure  p—  pf  equals  p"  to  cause  the  bending 
in  this  direction,  we  shall  have: 


But  as  all  these  strips  form  one  and  the  same  plate  the  ratio 
of  p'  to  p"  must  be  such  that  the  deflection  at  the  center  of  the 
plate  may  be  the  same  on  either  supposition.  The  general  for- 
mula for  deflection  in  this  case  is 

Wl* 
~         El 


84  MACHINE  DESIGN 

t3 
and   /  =  —  for   each   set   of   strips.     Therefore  the  deflection 

7>'Z4          v"b*  . 

is  proportional  to  *~-  and      3    in  the  two  cases. 
t  t 


But 

Solving  in  these  equations  for  p'  and  p" 


/Y\*    —  — 

p  ~ 


Substituting  these  values  in  (a)  and  (b) : 

(32) 

(33) 

As  l>b  usually,  equation  (33)  is  the  one  to  be  used.     If  the 
plate  is  square  l  =  b  and 

'=^/|  (34) 

If  the  plate  is  merely  supported  at  the  edges  then  formulas 
(32)  and  (33)  become: 
For  rectangular  plate : 


For  square  plate: 

(36) 


A  round  plate  may  be  treated  as  square,  with  side  —  diameter, 
without  sensible  error. 

The  preceding  formulas  can  only  be  regarded  as  approximate. 
Grashof  has  investigated  this  subject  and  developed  rational 
formulas  but  his  work  is  too  long  and  complicated  for  introduc- 
tion here.  His  formulas  for  round  plates  are  as  follows: 


FLAT  PLATES  85 

Round  plates: 
Supported  at  edges: 

/ '      I^P  fQ.'7\ 

~2\6S 
Fixed  at  edges: 


where  t  and  p  are  the  same  as  before,  d  is  the  diameter  in  inches 
and  S  is  the  safe  tensile  strength  of  the  material. 

Comparing  these  formulas  with  (34)  and  (36)  for  square  plates, 
they  are  seen  to  be  nearly  identical  if  allowance  is  made  for  the 
difference  in  the  value  of  S. 

Experiments  made  at  the  Case  School  of  Applied  Science  in 
1896-97  on  rectangular  cast-iron  plates  with  load  concentrated 
at  the  center  gave  results  as  follows:  Twelve  rectangular  plates 
planed  on  one  side  and  each  having  an  unsupported  area  of  10  by 
15  in.  were  broken  by  the  application  of  a  circular  steel  plunger 
1  in.  in  diameter  at  the  geometrical  center  of  each  plate.  The 
plates  varied  in  thickness  from  J  in.  to  1  J  in.  Numbers  1  to  6 
were  merely  supported  at  the  edges,  while  the  remaining  six  were 
clamped  rigidly  at  regular  intervals  around  the  edge. 

To  determine  the  value  of  S,  the  modulus  of  rupture  of  the 
mateiial,  pieces  were  cut  from  the  edge  of  the  plates  and  tested 
by  cross-breaking.  The  average  value  of  S  from  seven  experi- 
ments was  found  to  be  33,000  Ib.  per  square  inch. 

In  Table  XXV  are  given  the  values  obtained  for  the  breaking 
load  W  under  the  different  conditions. 

If  we  assume  an  empirical  formula: 


and  substitute  given  values  of  S,  I  and  6  we  have  nearly: 

w  =  mw.  (b) 

Substituting  values  of  W  and  t  from  the  Table  XXI  we  have 
the  values  of  k  as  given  in  the  last  column. 

If  we  average  the  values  for  the  two  classes  of  plates  and 
substitute  in  (a)  we  get  the  following  empirical  formulas: 


86 


MACHINE  DESIGN 


For  breaking  load  on  plates  supported  at  the  edges  and  loaded 
at  the  center: 

^  (39) 


= 
and  for  similar  plates  with  edges  fixed: 

tf/2 
TF-442 

*P+b* 

S  in  both  formulas  is  the  modulus  of  rupture. 

TABLE  XXV 

CAST-IRON  PLATES  10X15  IN. 


(40) 


No. 

Thickness  t 

Breaking  load  W 

Constant  k 

1 

.562 

7,500 

237 

2 

.641 

11,840 

288 

3 

.745 

14,800 

267 

4 

.828 

21,900 

320 

5 

1.040 

31,200 

289 

6 

1.120 

31,800 

254 

7 

.481 

9,800 

424 

8 

.646 

17,650 

422 

9 

.769 

26,400 

446 

10 

.881 

33,400 

430 

11 

1.020 

47,200 

454 

12                        1.123                           59,600 

477 

Those  plates  which  were  merely  supported  at  the  edges  broke 
in  three  or  four  straight  lines  radiating  from  the  center.  Those 
fixed  at  the  edges  broke  in  four  or  five  radial  lines  meeting  an 
irregular  oval  inscribed  in  the  rectangle.  Number  12,  however, 
failed  by  shearing,  the  circular  plunger  making  a  circular  hole  in 
the  plate  with  several  radial  cracks. 

Some  tests  were  made  in  the  spring  of  1906  at  the  Case  School 
laboratories  by  Messrs.  Hill  and  Nadig  on  the  strength  of  flat 
cast-iron  plates  under  uniform  hydraulic  pressure. 

Table  XXVI  gives  the  results  of  the  investigation. 

The  low  value  of  S  is  explained  by  the  fact  that  the  material 
was  a  soft  rather  coarse  gray  iron,  having  an  average  tensile 
strength  of  about  12,000  Ib. 


FLAT  PLATES 


87 


TABLE  XXVI 
CAST-IRON  PLATES,    UNIFORM  LOAD,  FIXED  EDGES 


Breaking  load  in  pounds  per 

Size  of  plate, 

Thickness 

Modulus 

square  inch 

inches 

Inches 

S 

By  formula 

Actual 

12X12 

0.75 

20,440 

(34)      320 

375 

12X12 

1.00 

27,900 

(34)      777 

675 

12X18 

0.94 

26,600 

(33)      390 

450 

12X18 

1.25 

24,000 

(33)      622 

650 

Further  experiments  are  needed  to  establish  any  general 
conclusions. 

32.  Steel  Plates.— Mr.  T.  A.  Bryson  of  Rensselaer  Polytechnic 
Institute  has  recently  made  some  tests  on  steel  plates  under 
hydrostatic  pressure  and  published  a  monograph  on  the  subject. 

The  material  tested  was  medium  steel  boiler  plate  from  J  to  \ 
in.  thick  and  the  sizes  used  were  18  by  18  in.  and  24  by  24  in. 

Two  plates  separated  by  a  cast-iron  distance  piece  were 
clamped  at  the  edges  by  cast-iron  frames  bolted  together. 
Hydrostatic  pressures  from  0  to  225  Ib.  per  square  inch  were 
applied  and  deflections  were  measured  at  five  points.  Both 
working  deflections  and  permanent  sets  were  noted.  The 
characteristics  of  the  material  were  determined  from  test  pieces 
cut  off  the  edge  of  each  plate. 

Mr.  Bryson  develops  formulas  similar  to  Morley's,1  which 
differ  from  those  just  given  in  the  values  of  the  constants.  All 
the  formulas  for  square  plates  can,  however,  be  reduced  to  the 
general  form : 

t  =  bjkj-  (See  formula  34) 

\  o 

or 

S  =  k*£  (41) 

where  S  is  the  maximum  stress  in  the  plate. 

The  value  of  k,  as  determined  by  the  average  of  eight  tests 
1  Morley's  Strength  of  Materials. 


88  MACHINE  DESIGN 

with  different  values  of  b  and  t,  was  0.141  at  the  elastic  limit  of 
the  material,  the  maximum  value  being  0.156  and  the  mini- 
mum 0.131. 

This  value  of  k  may  then  be  used  for  steel  plates  with  fixed 
edges  without  serious  error.  Mr.  Bryson  after  discussing  the 
experiments  of  Bach  on  square  and  rectangular  plates  recom- 
mends the  following  general  formula  for  rectangular  steel  plates 
fixed  at  the  edges  and  uniformly  loaded: 

0.5       .  Vp 
~  *~ 


where 

r  =  b/l 
Where  l  =  b  this  reduces  to  formula  (41). 

The  value  of  S  for  plates  merely  supported  may  be  assumed  to 
be  50  per  cent  greater  than  in  formula  (42)  . 

The  value  of  k  in  formula  (32)  is  determined  by  substituting 

r  =     and  reducing: 


.  r2 

2(1  +  r4) 
or  for  a  square  plate: 

«-J£  <*> 

These  values  of  k  are  much  larger  than  those  just  given.  In 
Mr.  Bryson'  s  tests  it  was  found  that  suspension  stresses  gradually 
supplanted  those  due  to  bending  and  that  this  change  reduced 
the  value  of  k. 

This  would  not  be  true  of  cast-iron  plates  and  the  formulas 
given  on  page  84  would  be  preferable. 

The  values  of  k  for  the  four  experiments  detailed  in  Table 
XXVI  would  be  respectively: 

fc=.213-.287-.363-.400 

which  shows  that  formula  (42)   is  not  applicable  to  cast-iron 
plates. 

The  most  comprehensive  experiments  on  flat  plates  are  those 
by  Professor  Bach,  and  Grashof's  formulas  are  largely  controlled 


FLAT  PLATES 


89 


by  them.1  Table  XXVII  gives  the  derived  formulas  for  some 
of  the  more  usual  cases.  The  notation  is  the  same  as  that  of 
the  previous  formulas. 

The  strength  of  the  plates  depends  also  on  the  manner  of 
fastening  at  the  edges,  the  number  and  size  of  bolts,  the  nature 
of  gasket  used,  if  any,  etc.,  etc. 

TABLE  XXVII 

STRESSES  IN  FLAT  PLATES 


Shape 

Edges 

Load 

Value  of 
fiber  stress 
S  = 

Value  of 
coefficient 
fc- 

Remarks 

Circle 

Fixed     .  . 

Uniform  .... 

pr* 

Cast  iron,  0.8  

r  =  radius. 

k~w 

Steel,  0.5  

Circle 

.  pr2 

Cast  iron    1  2 

T  =  radius. 

k-p 

Steel  0  7 

Ellipse.... 

Fixed  

Uniform.  .  .  . 

pb*          i 
Klt*(l  +  n*) 

Cast  iron,  1.34  
Steel   0  84 

Estimated. 

Ellipse.... 

Support.  .  . 

Uniform  .... 

pb*        i 
4£2(Z  4-  n2) 

Cast  iron,  2.26  
Steel,  1.41  

Estimated. 

Rect  

Fixed  

At  center.  .  . 

Wlb 

t2(l*+b*) 

Cast  iron,  2.63  

Rect 

Wlb 

Cast  iron    3  0 

kt*(i*+b*) 

Rect  

Fixed  

Uniform  .... 

k  -*l-b— 

m  i*+  62) 

Cast  iron,  0.38  
Steel   0  24 

Estimated. 

Rect  

Support.  .  . 

Uniform  .... 

PZ2&2 
P(l2+b2) 

Cast  iron,  0.57  
Steel,  0.36  

Estimated. 

Fixed 

At  center.  .  . 

.W 

Cast  iron,  1.32  

kT* 

Square..  .  . 

Support.  .  . 

At  center.  .  . 

k^ 

«  p 

Cast  iron,  1.50  



Square..  .  . 

Fixed  

Uniform  .... 

kptf 

Cast  iron,  0.19  
Steel,  0.12  

Estimated. 

Square.  .  .  . 

Support.  .  . 

Uniform  .... 

4° 

Cast  iron,  0.28  
Steel,  0.18  

Estimated. 

NOTE. — n 


minor  axis 


major  axis 

1  See  Am.  Mach.,  Nov.  25,  1909. 


90  MACHINE  DESIGN 

It  will  be  interesting  to  compare  values  of  S  in  Table  XXVII 
with  those  obtained  by  experiment  so  as  to  determine  whether 
S  corresponds  to  the  tensile  strength  of  the  metal  or  to  the 
modulus  of  rupture  in  cross  breaking. 

PROBLEMS 

1.  Calculate  the  thickness  of  a  steam-chest  cover  12X16  in.  to  sustain 
a  pressure  of  90  Ib.  per  square  inch  with  a  factor  of  safety  =  10. 

2.  Calculate  the  thickness  of  a  circular  manhole  cover  of  cast  iron  18  in. 
in  diameter  to  sustain  a  pressure  of  200  Ib.  per  square  inch  with  a  factor  of 
safety  =  8,  regarding  the  edges  as  merely  supported. 

3.  Determine  the  probable  breaking  load  for  a  plate  18X24  in.  loaded 
at  the  center,  (a)  when  edges  are  fixed.     (6)  When  edges  are  supported. 

4.  In  experiments  on  steam  cylinders,  a  head  12  in.  in  diameter  and  1.18 
in.  thick  failed  under  a  pressure  of  900  Ib.  per  square  inch.     Determine  the 
value  of  S  by  formula  (34). 

REFERENCES 

Mechanics  of  Materials,  Merriman,  Chapter  XIV. 

Strength  of  Materials,  Slocum  and  Hancock,  Chapters  VII  and  VIII. 

Details  of  High-pressure  Piping.     Cass.  June,  1906. 

Design  and  Construction  of  Piping.     Eng.  Mag.,  April,  1908. 

Piping  for  High  Pressures.     Power,  Sept.  22,  1908. 

Flanges  for  High  Pressures.     Power,  July,  1905;  Dec.,  1905. 

High-pressure  Tests  of  Large  Pipes.     Eng.  News,  Apr.  15,  1909. 


CHAPTER  IV 
FASTENINGS 

33.  Bolts  and  Nuts. — Tables  of  dimensions  for  U.  S.  standard 
bolt  heads  and  nuts  are  to  be  found  in  most  engineering  hand- 
books and  will  not  be  repeated  here. 

These  proportions  have  not  been  generally  adopted  on  account 
of  the  odd  sizes  of  bar  required.  The  standard  screw-thread  has 
been  quite  generally  accepted  as  superior  to  the  old  V-thread. 

Roughly  the  diameter  at  root  of  thread  is  0.83  of  the  outer 
diameter  in  this  system,  and  the  pitch  in  inches  is  given  by  the 
formula 


(45) 


where  d  =  outer  diameter. 


TABLE  XXVIII 

SAFE  WORKING  STRENGTH  OF  IRON  OR  STEEL  BOLTS 


Safe  load  in  tension, 

Safe  load  in  shear, 

Diam. 
of  bolt, 
inch 

Threads 
per  inch, 

No. 

Diam.  at 
root  of 
thread, 
inches 

Area  at 
root  of 
thread, 
sq.  in. 

pounds 

pounds 

5,000  Ib. 

7,500  Ib. 

4,000  Ib. 

6,000  Ib. 

per  sq.  in. 

per  sq.  in. 

per  sq.  in. 

per  sq.  in. 

i 

20 

.185 

.0269 

135 

202 

196 

294 

A 

18 

.240 

.0452 

226 

340 

306 

.  460 

i 

16 

.294 

.0679 

340 

510 

440 

660 

i7s 

14 

.344 

.0930 

465 

695 

600 

900 

* 

13 

.400 

.1257 

628 

940 

785 

1,175 

91 


92 


MACHINE  DESIGN 


TABLE  XXVIII     (Continued) 
SAFE  WORKING  STRENGTH  OF  IRON  OR  STEEL  BOLTS 


Safe  load  in  tension, 

Safe  load  in  shear, 

Diam. 
of  bolt, 

•          i_ 

Threads 
per  inch, 

XT 

Diam  at 
root  of 
thread, 

Area  at 
root  of 
thread, 

pounds 

pounds 

| 

men 

.No. 

inches 

sq.  in. 

5,000  Ib. 

7,500  Ib. 

4,000  Ib. 

6,000  Ib. 

per  sq.  in. 

per  sq.  in. 

per  sq.  in. 

per  sq.  in. 

A 

12 

.454 

.162 

810 

1,210 

990 

1,485 

1 

11 

.507 

.202 

1,010 

1,510 

1,230 

1,845 

i 

10 

.620 

.302 

1,510 

2,260 

1,770 

2,650 

I 

9 

.731 

.420 

2,100 

3,150 

2,400 

3,600 

i 

8 

.837 

.550 

2,750 

4,120 

3,140 

4,700 

H 

7 

.940 

.694 

3,470 

5,200 

3,990 

6,000 

U 

7 

1.065 

.891 

4,450 

6,680 

4,910 

7,360 

if 

6 

1.160 

1.057 

5,280 

7,920 

5,920 

7,880 

H 

6 

1.284 

1.295 

6,475 

9,710 

7,070 

10,600 

If 

54 

1.389 

1.515 

7,575 

11,350 

8,250 

12,375 

if 

5 

1.490 

1.744 

8,720 

13,100 

9,630 

14,400 

U 

5 

1.615 

2.049 

10,250 

15,400 

11,000 

16,500 

2 

4i 

1.712 

2.302 

11,510 

17,250 

12,550 

18,800 

The  shearing  load  is  calculated  fiom  the  area  of  the  body  of 
the  bolt. 

Bolts  may  be  divided  into  three  classes  which  are  given  in  the 
order  of  their  merit. 

1.  Through  bolts,  having  a  head  on  one  end  and  a  nut  on  the 
other. 

2.  Stud  bolts,  having  a  nut  on  one  end  and  the  other  screwed 
into  the  casting. 

3.  Tap  bolts  or  screws  having  a  head  at  one  end  and  the  other 
screwed  into  the  casting. 

The  principal  objection  to  the  last  two  forms  and  especially  to 
(3)  is  the  liability  of  sticking  or  breaking  off  in  the  casting. 

Any  irregularity  in  the  bearing  sui faces  of  head  or  nut  where 
they  come  in  contact  with  the  casting,  causes  a  bending  action 
and  consequent  danger  of  rupture. 

This  is  best  avoided  by  having  a  slight  annular  projection  on 
the  casting  concentric  with  the  bolt  hole  and  finishing  the  flat 
surface  by  planing  or  counter-boring. 

Counter-boring  without  the  projection  is  a  rather  slovenly  way 
of  over  coming  the  difficulty. 


BOLTS  AND  NUTS  93 

When  bolts  or  studs  are  subjected  to  severe  stress  and  vibration, 
it  is  well  to  turn  down  the  body  of  the  bolt  to  the  diameter  at 
root  of  thread,  as  the  whole  bolt  will  then  stretch  slightly  under 
the  load. 

A  check  nut  is  a  thin  nut  screwed  firmly  against  the  main  nut 
to  prevent  its  working  loose,  and  is  commonly  placed  outside. 

As  the  whole  load  is  liable  to  come  on  the  outer  nut,  it  would 
be  more  correct  to  put  the  main  nut  outside.  (Prove  this  by 
figure.) 

After  both  nuts  aie  firmly  screwed  down,  the  outer  one  should 
be  held  stationary  and  the  inner  one  reversed  against  it  with 
what  force  is  deemed  safe,  that  the  greater  reaction  may  be 
between  the  nuts. 

Numerous  devices  have  been  invented  for  the  purpose  of  hold- 
ing nuts  from  working  loose  under  vibration  but  none  of  them  are 
entirely  satisfactory. 

Probably  the  best  method  for  large 
nuts  is  to  drive  a  pin  or  cotter  entirely 
through  nut  and  bolt. 

A  flat  plate,  cut  out  to  embrace  the 
nut  and  fastened  to  the  casting  by  a 
machine  screw,  is  often  Used. 

Machine  Screws. — A  screw  is  distin- 
guished from  a  bolt  by  having  a  slot- 
ted, round  head  instead  of  a  square  or 
hexagon  head. 

The  head  may  have  any  one  of  four  FIG>  26. 

shapes,  the  round,  fillister,  oval  fillister 

and  flat  as  shown  in  Fig.  26.  A  committee  of  the  American 
Society  of  Mechanical  Engineers  has  recently  recommended 
certain  standards  for  machine  screws.  The  form  of  thread 
recommended  is  the  U.  S.  Standard  or  Sellers  type  with  provi- 
sion for  clearance  at  top  and  bottom  to  insure  bearing  on  the 
body  of  the  thread. 

The  sizes  and  pitches  recommended  are  shown  in  Table  XXIX. 

In  designing  eye-bolts  it  is  customary  to  make  the  combined 
sectional  area  of  the  sides  of  the  eye  one  and  one-half-times  that 
of  the  bolt  to  allow  for  obliquity  and  an  uneven  distribution  of 
stress. 


94 


MACHINE  DESIGN 


TABLE  XXIX 

MACHINE  SCREWS 


Standard  diam. 

.070 

.085 

.100 

.110 

.125 

.140 

.165 

.190 

.215 

.240 

.250 

.270 

.320 

.375 

Threads  per  in. 

72 

64 

56 

48 

44 

40 

36 

32 

28 

24 

24 

22 

20 

16 

Reference  is  made  to  the  report  itself  for  further  details  of 
heads,  taps,  etc. 

34.  Crane  Hooks. — Heretofore,  the  large  wrought-iron  or  steel 
hooks  used  for  crane  service  have  usually  been  designed  by  con- 
sidering the  fibers  on  the  inside  of  a  hook  to  be  subjected  to  a 
tension  which  was  the  resultant  of  the  direct  load  and  of  the 
bending  due  to  the  eccentricity  of  the  loading. 

Experiments  made  by  Professor  Rautenstrauch  in  19091 
show  that  such  methods  do  not  give  correct  results.  Ten  hooks 
of  various  capacities  were  tested  by  direct  loading  and  their 
elastic  limits  determined. 

The  following  table  gives  the  leading  data  and  results.  The 
dimensions  are  those  of  the  principal  cross-section: 

TABLE  XXX 

ELASTIC  LIMIT  OF  CRANE  HOOKS 


Nominal 

Cross-section  dimensions 

Elastic 

capacity, 

Material 

limit, 

tons 

Ib. 

A 

/ 

I 

y 

30 

C.  steel  .  .  . 

23.35 

111.6 

7.25 

3.36 

56,000 

20 

C.  steel  .  .  . 

14.48 

5.90 

2.75 

30,000 

15 

C.  steel  .  .  . 

13.92 



5.13 

2.23 

48,000 

15 

W.  iron... 

8.40 

11.9 

5.00 

1.87 

16,000 

10 

C.  steel  .  .  . 

8.72 

4.30 

2.05 

18,000 

10 

W.  iron... 

6.08 

6.5 

4.00 

1.50 

16,000 

5 

C.  steel.  .  . 

5.69 

3.25 

1.42 

18,000 

5 

W.  iron  .  .  . 

4.80 

3.8 

3.47 

1.35 

14,000 

3 

C.  steel... 

3.50 



2.89 

1.16 

8,500 

2 

C.  steel.  .  . 

2.03 

2.03 

0.88 

4,700 

1  Am.  Mach.,  Oct.  7,  1909. 


CRANE  HOOKS 


95 


A  —  area  in  square  inches 
1  =  moment  of  inertia  about  gravity  axis 
1=  distance  from  load  line  to  gravity  axis 
y  =  distance  from  inner  fiber  to  gravity  axis. 

It  will  be  noticed  that  the  nominal  capacity  of  the  hook  is  in 
several  cases  greater  than  the  elastic  limit  as  shown  by  experi- 
ment. This  is  particularly  true  ot  the  larger  sizes. 

The  standard  cross-section  of  crane  hooks  is  that  of  a  trapezoid 
with  curved  bases  as  shown  in  Fig.  27.  The  wider  base  corre- 
sponds to  the  inner  side  of  the  hook  where 
the  tension  is  greatest. 

The  dimensions  given  are  approxi- 
mately those  of  a  20-ton  steel  hook. 
Professor  Rautenstrauch  finds  that  the 
values  of  the  load  at  elastic  limit,  as  de- 
termined by  the  ordinary  formula  above 
alluded  to,  are  entirely  erroneous,  being 
in  many  cases  more  than  twice  that  found 
by  the  actual  tests.  He  recommends  in- 
stead the  so-called  Andrews-Pearson  for- 
mula which  takes  into  account  the  curva- 
ture of  the  neutral  axis  and  the  lateral 
distortion  of  the  metal. 

The  discussion  is  too  long  for  reproduc- 
tion here  and  reference  is  made  to  his  paper 
and  to  the  original  presentation  of  this 
formula.1 

A  similar  condition  exists  in  large  chain  links.  The  bending 
moment  in  this  case  is,  however,  usually  eliminated  by  the 
insertion  of  a  cross  piece  or  strut.2 

PROBLEMS 

1.  Discuss  the  effect  of  the  initial  tension  caused  by  the  screwing  up  of 
the  nut  on  the  bolt,  in  the  case  of  steam  fittings,  etc.;  i.e.,  should  this  tension 
be  added  to  the  tension  due  to  the  steam  pressure,  in  determining  the  proper 
size  of  bolt? 

1  Technical  Series  1,  Draper  Company's  Research  Memoirs,  1904.     See 
also  Slocum  and  Hancock's  Strength  of  Materials. 

2  See  Univ.  of  Illinois,  Bulletin  No.  18.       "  The  Strength  of  Chain  Links," 
by  G.  A.  Goodenough  and  L.  E.  Moore. 


3.25-4 


FIG.  27.— 20-ton  steel 
hook. 


96 


MACHINE  DESIGN 


i 

II 


2.  Determine  the  number  of  J-in.  steel  bolts  necessary  to  hold  on  the 
head  of  a  steam  cylinder  18  in.  diameter,  with  the  internal  pressure  90  Ib. 
per  square  inch,  and  factor  of  safety  =  12. 

3.  Show  what  is  the  proper  angle  between  the  handle  and  the  jaws  of  a 
fork  wrench. 

(1)  If  used  for  square  nuts. 

(2)  If  used  for  hexagon  nuts;  illustrate  by  figure. 

4.  Determine  the  length  of  nut  theoretically  necessary  to  prevent  stripping 
of  the  thread,  in  terms  of  the  outer  diameter  of  the  bolt. 

(1)  With  U.  S.  standard  thread. 

(2)  With  square  thread  of  same  depth. 

5.  Design  a  hook  with  a  swivel  and  eye  at  the  top  to  hold  a  load  of  10 
tons  with  a  factor  of  safety  5,  the  center  line  of  hook  being  8  in.  from  line  of 
load,  and  the  material  soft  steel. 

35.  Riveted  Joints. — Riveted  joints  may  be  divided  into  two 
general  classes:  lap  joints  where  the  two  plates  lap  over  each 

other,  and  butt  joints  where 
the  edges  of  the  plates  butt 
_„__  together  and  are  joined  by 

j         A Q Oi^    over-lapping  straps  or  welts. 

If  the  strap  is  on  one  side 
only,  the  joint  is  known  as  a 
butt  joint  with  one  strap:  if 
straps  are  used  inside  and 
out  the  joint  is  called  a  butt 
joint  with  two  straps.  Butt 
joints  are  generally  used  when  the  material  is  more  than  \ 
in.  thick. 

Any  joint  may  have  one, 
two  or  more  rows  of  rivets 
and  hence  be  known  as  a 
single  riveted  joint,  a  double 
riveted  joint,  etc. 

Any  riveted  joint  is  weaker 
than  the  original  plate,  simply 
because  holes  cannot  be 
punched  or  drilled  in  the 
plate  for  the  introduction  of 
rivets  without  removing  some  of  the  metal. 

Fig.  28  shows  a  double  riveted  lap  joint  and  Fig.  29  a  single 
riveted  butt  joint  with  two  straps. 


^^m 

^  %l^SS 


<;yj^  -^ 
%f^  S 


FIG.  28. 


c 


c 


FIG.  29. 


RIVETED  JOINTS 


97 


Riveted  joints  may  fail  in  any  one  of  four  ways:  • 

1.  By  tearing  of  the  plate  along  a  line  of  rivet  holes,  as  at  A  B, 
Fig.  28. 

2.  By  shearing  of  the  rivets. 

3.  By  crushing  and  wrinkling  of  the  plate  in  front  of  each  rivet 
as  at  C,  Fig.  28,  thus  causing  leakage. 

4.  By  splitting  of  the  plate  opposite  each  rivet  as  at  D,  Fig.  28. 
The  last  manner  of  failure  may  be  prevented  by  having  a  suffi- 
cient distance  from  the  rivet  to  the  edge  of  the  plate.     Practice 
has  shown  that  this  distance  should  be  at  least  equal  to  the 
diameter  of  a  rivet. 

Experience  has  shown  that  lap  joints  in  plates  of  even  moderate 
thickness  are  dangerous  on  account  of  the  liability  of  hidden 
cracks.  Several  disastrous  boiler  explosions  have  resulted  from 
the  presence  of  cracks  inside  the  joint  which  could  not  be  detected 
by  inspection.  The  fact  that  one  or  both  plates  are  out  of  the 
line  of  pull  brings  a  bending  moment  on  both  plates  and  rivets. 

Some  boiler  inspectors  have  gone  so  far  as  to  condemn  lap 
joints  altogether. 

Let         £  =  thickness  of  plate 

d  =  diameter  of  rivet  hole 

p  =  pitch  of  rivets 

n  =  number  of  rows  of  rivets 

T  =  tensile  strength  of  plate 

C  =  crushing  strength  of  plate  or  rivet 

S=  shearing  strength  of  rivet. 
Average  values  of  the  constants  are  as  follows: 


Material 

T 

C 

s 

Wrought  iron  
Soft  steel 

50,000 
56000 

80,000 
90000 

40,000 
45  000 

The  values  of  the  constants  given  above  are  only  average 
values  and  are  liable  to  be  modified  by  the  exact  grade  of  material 
used  and  the  manner  in  which  it  is  used. 

The  tensile  strength  of  soft  steel  is  reduced  by  punching  from 


98  MACHINE  DESIGN 

3  to  12  per  cent  according  to  the  kind  of  punch  used  and  the 
width  of  pitch.  The  shearing  strength  of  the  rivets  is  diminished 
by  their  tendency  to  tip  over  or  bend  if  they  do  not  fill  the  holes, 
while  the  bearing  or  compression  is  doubtless  relieved  to  some 
extent  by  the  friction  of  the  joint.  The  values  given  allow 
roughly  for  these  modifications. 

36.  Lap  Joints.  —  This  division  also  includes  butt  joints  which 
have  but  one  strap. 

Let  us  consider  the  shell  as  divided  into  strips  at  right  angles 
to  the  seam  and  each  of  a  width  =  p.  Then  the  forces  acting  on 
each  strip  are  the  same  and  we  need  to  consider  but  one  strip. 

The  resistance  to  tearing  across  of  the  strip  between  rivet  holes 

is  (p-d)tT  (a) 

and  this  is  independent  of  the  number  of  rows  of  rivets. 
The  resistance  to  compression  in  front  of  rivets  is 

ndtC  (b) 

and  the  resistance  to  shearing  of  the  rivets  is 

^nd'S.  (c) 

If  we  call  the  tensile  strength  T  =  unity  then  the  relative 
values  of  C  and  S  are  1.6  and  0.8  respectively. 

Substituting  these  relative  values  of  T,  C  and  S  in  our  equations, 
by  equating  (b)  and  (c)  and  reducing  we  have 

d  =  2.55Z  (46) 

Equating  (a)  and  (c)  and  reducing  we  have 


(47) 
t 

Or  by  equating  (a)  and  (b) 

p  =  d  +  l.Gnd  (48) 

These  proportions  will  give  a  joint  of  equal  strength  throughout, 
for  the  values  of  constants  assumed. 

37.  Butt  Joints  with  Two  Straps.  —  In  this  case  the  resistance  to 
shearing  is  increased  by  the  fact  that  the  rivets  must  be  sheared 


RIVETED  JOINTS  99 

at  both  ends  before  the  joint  will  fail.     Experiment  has  shown 
this  increase  of  shearing  strength  to  be  about  85  per  cent  and  we 
can  therefore  take  the  relative  value  of  S  as  1.5  for  butt  joints. 
This  gives  the  following  values  for  d  and  p 

d  =  lMt  (49) 

t?r/2 
p  =  d  +  1.18—  (50) 

6 

p  =  d  +  l.6nd.  .  (51) 

In  the  preceding  formulas  the  diameter  of  hole  and  rivet  have 
been  assumed  to  be  the  same. 

The  diameter  of  the  cold  rivet  before  insertion  will  be  y1^  in. 
less  than  the  diameter  given  by  the  formulas. 

Experiments  made  in  England  by  Prof.  Kennedy  give  the 
following  as  the  proportions  of  maximum  strength : 

Lap  joints  d  =  2.33t 

p  =  d  + 

Butt  joints  d  =  l.St 


38.  Efficiency  of  Joints.  —  The  efficiency  of  joints  designed  like 
the  preceding  is  simply  the  ratio  of  the  section  of  plate  left 
between  the  rivets  to  the  section  of  solid  plate,  or  the  ratio  of  the 
clear  distance  between  two  adjacent  rivet  holes  to  the  pitch. 
From  formula  (48)  we  thus  have: 

Efficiency  =  .  (52) 


This  gives  the  efficiency  of  single,  double  and  triple  riveted 
seams  as 

.615,  .762  and  .828  respectively. 

Notice  that  the  advantage  of  a  double  or  triple  riveted  seam 
over  the  single  is  in  the  fact  that  the  pitch  bears  a  greater  ratio 
to  the  diameter  of  a  rivet,  and  therefore  the  proportion  of  metal 
removed  is  less. 


100 


MACHINE  DESIGN 


39.  Butt  Joints  with  Unequal  Straps.  —  One  joint  in  common  use 
requires  special  treatment. 

It  is  a  double  riveted  butt  joint  in  which  the  inner  strap  is 
made  wider  than  the  outer  and  an  extra  row  of  rivets  added, 
whose  pitch  is  double  that  of  the  original  seam;  this  is  sometimes 
called  diamond  riveting.  See  Fig.  30. 

This  outer  row  of  rivets  is 
then  exposed  to  single  shear 
and  the  original  rows  to  dou- 
ble shear. 

Consider  a  strip  of  plate  of 
a  width  =  2p.  Then  the  resis- 
tance to  tearing  along  the 
outer  row  of  rivets  is 

(2p-d)tT 

As  there  are  five  rivets  to 
compress  in  this  strip  the 
bearing  resistance  is 

__  5dtC 

As    there    is  one   rivet    in 
single  shear  and  four  in  double  shear  the  resistance  to  shearing  is 

+  (4X1.85)   |    7t4d2S  =  QM2S 

Solving  these  equations  as  in  previous  cases,  we  have  for  this 
particular  joint 

d  =  1.52t  (53) 


1 

1 

> 

} 

1  1 
0          0 

1 

) 

o  o  o' 

ii 

o  o  o 

1- 

o  o  o 

it 

)" 

o  o  o 

1 

^> 

1 

o       o 

1 

J 

J 

iHI 

Vffi  .  2p-d      8 

Efficiency  ==-^—=  - 


(54) 
(55) 


40.  Practical  Rules. — The  formulas  given  above  show  the 
proportions  of  the  usual  forms  of  joints  for  uniform  strength. 

In  practice  certain  modifications  are  made  for  economic  reasons. 
To  avoid  great  variation  in  the  sizes  of  rivets  the  latter  are  graded 
by  sixteenths  of  an  inch,  making  those  for  the  thicker  plates  con- 


RIVETED  JOINTS 


101 


siderably  smaller  than  the  formula  would  allow,  and  the  pitch  is 
then  calculated  to  give  equal  tearing  and  shearing  strength. 

Table  XXXI  shows  what  may  be  considered  average  practice 
in  this  country  for  lap  joints  with  steel  plates  and  rivets. 


TABLE  XXXI 
RIVETED  LAP  JOINTS 


Thick- 

Diam. 

Diam. 

Pitch 

Efficiency  of  plate 

ness  of 

of 

of 

plate 

rivet 

hole 

Single 

Double 

Single 

Double 

t 

* 

A 

If 

If 

.59 

.68 

A 

f 

H 

If 

2* 

.58 

.68 

I 

1 

« 

If 

2J 

.57 

.67 

T7ff 

it 

1 

2 

2f 

.56 

.68 

i 

1 

if 

2 

2f 

.53 

.67 

The  efficiencies  are  calculated  from  the  strength  of  plate 
between  rivet  holes  and  the  efficiencies  of  the  rivets  may  be  even 
lower.  Comparing  these  values  with  the  ones  given  in  Art.  38 
we  find  them  low.  This  is  due  to  the  fact  that  the  pitches 
assumed  are  too  small.  The  only  excuse  for  this  is  the  possibility 
of  getting  a  tighter  joint. 

TABLE  XXXII 
RIVETED  BUTT  JOINTS 


Pitch 

Thickness  of 

Diam.  of 

Diam.  of 

plate 

rivet 

hole 

Single 

Double 

Triple 

t 

f 

U 

2f 

4 

5i 

f 

13 
16 

'    1 

2| 

3| 

5t 

1 

1 

11 

2| 

3| 

6* 

1 

If 

1 

2f 

31 

5 

1 

1 

IA 

2f 

3f 

5 

102  MACHINE  DESIGN 

Table  XXXII  has  been  calculated  for  butt  joints  with  two 
straps.  As  in  the  preceding  table  the  values  of  the  pitch  are  too 
small  for  the  best  efficiency.  The  tables  are  only  intended  to 
illustrate  common  practice  and  not  to  serve  as  standards.  There 
is  such  a  diversity  of  practice 'among  manufacturers  that  it  is 
advisable  for  the  designer  to  proportion  each  joint  according  to 
his  own  judgment,  using  the  rules  of  Arts.  36-39  and  having 
regard  to  the  practical  considerations  which  have  been  mentioned. 

A  committee  of  the  Master  Steam  Boiler  Makers'  Association 
has  made  a  number  of  tests  on  riveted  joints  and  reported  its 
conclusions.  The  specimens  were  prepared  according  to  generally 
accepted  practice,  but  on  subjecting  them  to  tension  many  of 
them  failed  by  tearing  through  from  hole  to  edge  of  plate.  The 
committee  recommends  making  this  distance  greater,  so  that 
from  the  center  of  hole  to  edge  of  plate  shall  be  perhaps  2d  instead 
of  1.5d. 

The  committee  further  found  the  shearing  strength  of  rivets 
to  be  in  pounds  per  square  inch  of  section. 


Single  shear 

Double  shear 

Iron  rivets  
Steel  rivets 

40,000 
49  000 

78,000 
84  000 

Compare  these  values  with  those  given  in  Art.  35.  Also  note 
that  the  factor  for  double  shear  is  1.95  for  iron  rivets  and  only 
1.71  for  steel  rivets  as  against  the  1.85  given  in  Art.  37.  The 
committee  found  that  machine-driven  rivets  were  stronger  in 
double  shear  than  hand-driven  ones. 

PROBLEMS 

1.  Calculate  diameter  and  pitch  of  rivets  for  J-in.  and  £-in.  plate  and 
compare  results  with  those  in  Table  XXXI.     Criticise  latter. 

2.  Show  the  effect  in  Prob.  1  of  using  iron  rivets  in  steel  plates. 

3.  Criticise  proportions  of  joints  for  J-in.  and  1-in.  plate  in  Table  XXXII 
by  testing  the  efficiency  of  rivets  and  plates. 

4.  A  cylinder  boiler  5X16  ft.  is  to  have  long  seams  double  riveted  laps 
and  ring  seams  single  riveted  laps.     If  the  internal  pressure  is  90  Ib.  gage 
pressure  and  the  material  soft  steel,  determine  thickness  of  plate  and  pro- 
portion of  joints.     The  net  factor  of  safety  at  joints  to  be  5. 


RIVETED  JOINTS 


103 


5.  A  marine  boiler  is  13  ft.  6  in.  in  diameter  and  14  ft.  long*  The  long 
seams  are  to  be  diamond  riveted  butt  joints  and  the  ring  seams  ordinary 
double  riveted  butt  joints.  The  internal  pressure  is  to  be  175  Ib.  gage  and 
the  material  is  to  be  steel  of  60,000  Ib.  tensile  strength.  Determine  thick- 
ness of  shell  and  proportions  of  joints.  Net  factor  of  safety  to  be  5,  as  in 
Prob.  4. 


r\ 


^///////7////////////////////y//// 


r\ 


FIG.  31. 

6.  Design  a  diamond  riveted  joint  such  as  shown  in  Fig.  31  for  a  steel 
plate  f  in.  thick.     Outer  cover  plate  is  f  in.  and  inner  cover  plate  is  TV  in. 
thick;  the  pitch  of  outer  rows  of  rivets  to  be  twice  that  of  inner  rows. 
Determine  efficiency  of  joints. 

7.  The  single  lap  joint  with  cover  plate,  as  shown  in  Fig.  32,  is  to  have 
pitch  of  outer  rivets  double  that  of  inner  row.      Determine  diameter  and 
pitch  of  rivets  for  |-in.  plate  and  the  efficiency  of  joint. 


41.  Riveted  Joints  for  Narrow  Plates. — The  joints  which  have 
been  so  far  described  are  continuous  and  but  one  strip  of  a  width 
equal  to  the  pitch  or  the  least  common  multiple  of  several 
pitches,  has  been  investigated. 

When  narrow  plates  such  as  are  used  in  structural  work  are 
to  be  joined  by  rivets,  the  joint  is  designed  as  a  whole.  Diamond 
riveting  similar  to  that  shown  in  Fig.  30  is  generally  used  and  the 
joint  may  be  a  lap,  or  a  butt  with  double  straps.  The  diameter  of 
rivets  may  be  taken  about  1.5  times  the  thickness  of  plate  [see 
equation  (53)],  and  enough  rivets  used  so  that  the  total  shearing 
strength  may  equal  the  tensile  strength  of  the  plate  at  the  point 
of  the  diamond,  where  there  is  one  rivet  hole.  It  may  be  neces- 
sary to  put  in  more  rivets  of  a  less  diameter  in  order  to  make 
the  figure  symmetrical. 


104 


MACHINE  DESIGN 


The  efficiency  of  the  joint  may  be  tested  at  the  different  rows 
of  rivets,  allowing  for  tension  of  plate  and  shear  of  rivets  in  each 
case. 

PROBLEMS 

1.  Design  a  diamond  riveted  lap  joint  for  a  plate  12  in.  wide  and  f  in. 
thick,  and  calculate  least  efficiency  for  shear  and  tension. 

2.  A  diamond  riveted  butt  joint  with  two  straps  has  rivets  arranged  as  in 
Fig.  33,  the  plate  being  12  in.  wide  and  f  in.  thick,  and  the  rivets  being  1 
in.  in  diameter.     If  the  plate  and  rivets  are  of  steel,  find  the  probable 
ultimate  strength  of  the  following  parts: 

(a)  The  whole  plate. 

(6)  All  the  rivets  on  one  side  of  the  joint. 

(c)  The  joint  at  the  point  of  the  diamond. 

(d)  The  joint  at  the  row  of  rivets  next  the  point. 


o 
o  o 

000 

o  o  o 

o  o 

o 


FIG.  33. 


42.  Joint  Pins. — A  joint  pin  is  a  bolt  exposed 
to  double  shear.  If  the  pin  is  loose  in  its  bear- 
ings it  should  be  designed  with  allowance  for 
bending,  by  adding  from  30  to  50  per  cent  to  the 
area  of  cross-section  needed  to  resist  shearing 
alone.  Bending  of  the  pin  also  tends  to  spread 
apart  the  bearings  and  this  should  be  prevented 
by  having  a  head  and  nut  or  cotter  on  the  pin. 

If  the  pin  is  used  to  connect  a  knuckle  joint 
as  in  boiler  stays,  the  eyes  forming  the  joint 
should  have  a  net  area  50  per  cent  in  excess  of 

the  body  of  the  stay,  to  allow  for  bending  and  uneven  tension 

(see  Eye-bolts,  Art.  33). 
Fig.  34  shows  a  pin  and  angle  joint  for  attaching  the  end  of  a 

boiler  stay  to  the  head  of  the  boiler. 

43.  Cotters. — A  cotter  is  a  key  which  passes  diametrically 
through  a  hub  and  its  rod  or  shaft,  to  fasten  them  together,  and 
is  so  called  to  distinguish  it  from  shafting  keys  which  lie  parallel 
to  axis  of  shaft. 

Its  taper  should  not  be  more  than  4  degrees  or  about  1  in  15, 
unless  it  is  secured  by  a  screw  or  check  nut. 

The  rod  is  sometimes  enlarged  where  it  goes  in  the  hub,  so  that 
the  effective  area  of  cross-section  where  the  cotter  goes  through 
may  be  the  same  as  in  the  body  of  the  rod.  (See  Fig.  35.) 


COTTERS 


105 


Let:      d  =  diameter  of  body  of  rod 

d^  =  diameter  of  enlarged  portion 

t  —  thickness  of  cotter,  usually  =-£ 

b  =  breadth  of  cotter 
1  =  length  of  rod  beyond  cotter. 

Suppose  that  the  applied  force  is  a  pull  on  the  rod — causing 
tension  on  the  rod  and  shearing  stress  on  the  cotter. 
The  effective  area  of  cross-section  of  rod  at  cotter  is 


nan 


FZG.  34. 


FIG.  35. 


nd*     d*_  <V 

'-  } 


and  this  should  equal  the  area  of  cross-section  of  the  body  of  rod. 


(56) 


Let  P^pull  on  rod. 

S  =  shearing  strength  of  material. 
The  area  to  resist  shearing  of  cotter  is 


•   h       2P 
-d^' 

The  area  to  resist  shearing  of  rod  is 

M.I-J 


(a) 


106  MACHINE  DESIGN 

P 


If  the  metal  of  rod  and  cotter  are  the  same 


l  =   '  (57) 

Great  care  should  be  taken  in  fitting  cotters  that  they  may 
not  bear  on  corners  of  hole  and  thus  tear  the  rod  in  two. 

A  cotter  or  pin  subjected  to  alternate 
{     ^  stresses    in    opposite  directions   should 

I       |  —  ^  have    a    factor   of   safety   double   that 

f  |i          \      otherwise  allowed. 

!  I       Adjustable  cotters,  used  for  tighten- 

I  ing  joints  of  bearings  are  usually  ac- 
I  _  \  companied  by  a  gib  having  a  taper  equal 


v — I — 1 LL_ i     and  opposite  to  that  of  the  cotter  (Fig. 

| f 36).     In    designing   these  for  strength 

JTIG   35  the   two    can   be   regarded  as  resisting 

shear  together. 

For  shafting  keys  see  chapter  on  shafting. 
The  split  pin  is  in  the  nature  of  a  cotter  but  is  not  usually 
expected  to  take  any  shearing  stress. 

PROBLEMS 

1.  Design  an  angle  joint  for  a  soft  steel  boiler  stay,  the  pull  on  stay  being 
12,000  Ib..  and  the  factor  of  safety,  6.     Use  two  standard  angles. 

2.  Determine  the  diameter  of  a  round  cotter  pin  for  equal  strength  of 
rod  and  pin. 

3.  A  rod  of  wrought  iron  has  keyed  to  it  a  piston  24  in.  in  diameter,  by  a 
cotter  of  machinery  steel. 

Required  the  two  diameters  of  rod  and  dimensions  of  cotter  to  sustain  a 
pressure  of  150  Ib.  per  square  inch  on  the  piston.     Factor  of  safety  =  8. 

4.  Design  a  cotter  and  gib  for  connecting  rod  of  engine  mentioned  in 
Prob.  3,  both  to  be  of  machinery  steel  and  .75  in.  thick.     (See  Fig,  36.) 

REFERENCES 

Machine  Design.     Unwin.,  Vol.  I,  Chapters  IV  and  V. 
Failures  of  Lap  Joints.     Power,  May,  1905;  Feb.,  1907;  Nov.,  1907. 
Tests  of  Nickel  Steel  Riveted  Joints.     By  A.  N.  Talbot  and  H.  F.  Moore. 
University  o/  Illinois  Bulletin,  No.  49. 


CHAPTER  V 
SPRINGS 

44.  Helical  Springs.  —  The  most  common  form  of  spring  used 
in  machinery  is  the  spiral  or  helical  spring  made  of  round  brass 
or  steel  wire.  Such  springs  may  be  used  to  resist  extension  or 
compression  or  they  may  be  used  to  resist  a  twisting  moment. 

Tension  and  Compression 
Let  L  —  length  of  axis  of  spring 
D  =  mean  diameter  of  spring 
I  =  developed  length  of  wire 
d  =  diameter  of  wire 

R  =  ratio  -j 
a 

n  —  number  of  coils 

P  =  tensile  or  compressive  force 

x  =  corresponding  extension  or  compression 

S  =  safe  torsional  or  shearing  strength  of  wire 

=  45,000  to  60,000  for  spring  brass  wire 

=  75,000  to  115,000  for  cast  steel,  tempered 
G  =  modulus  of  torsional  elasticity 

=  6,000,000  for  spring  brass  wire 

=  12,000,000  to  15,000,000  for  cast  steel,  tempered. 


Then  I  = 

If  the  spring  were  extended  until  the  wire  became  straight  it 
would  then  be  twisted  n  times,  or  through  an  angle  =  2xn  and 
the  stretch  would  be  I  —  L. 

The  angle  of  torsion  for  a  stretch  =  x  is  then 


Suppose  that  a  force  Pf  acting  at  a  radius  -=  will  twist  this 

107 


108  MACHINE  DESIGN 

same  piece  of  wire  through  an  angle  0  causing  a  stress  S  at  the 

surface  of  the  wire.     Then  will  the  distortion  of  the  surface  of  the 

fi/i 
wire  per  inch  of  length  be  s  =  ^y  and  the  stress 

5.1  71     Z.IP'D 

S  =  :-^-:  ~2dT  (b) 

n     S     1Q.2P'DI 

"G  =  ~s=^W  (C) 

In  thus  twisting  the  wire  the  force  required  will  vary  uniformly 
from  o  at  the  beginning  to  Pf  at  the  end  provided  the  elastic 
limit  is  not  passed,  and  the  average  force  will  be 

Pf  P'DO 

=  -fr     The  work  done  is  therefore  - 
2  4 

If  the  wire  is  twisted  through  the  same  angle  by  the  gradual 
application  of  the  direct  pressure  P,  compressing  or  extending 
the  spring  the  amount  x,  the  work  done  will  be 

Px  P'DO     Px 

T 


fd) 

DO 

Substituting  this  value  of  P'  in  (c)  and  solving  for  x: 

_  GdW 
~  10.2PZ 

Substituting  the  value  of  0  from  (a)  and  again  solving  for  x: 
10.2PZ  ll- 


If  we  neglect  the  original  obliquity  of  the  wire  then  l  =  nDn 
and  L  —  o  and  equation  (e)  reduces  to 

2.55PZD2 
x=  -^T-  (58) 

Making  the  same  approximation  in  equation   (d)   we  have 

P'=P 

i.e.  —  a  force  P  will  twist  the  wire  through  approximately  the 
same  angle  when  applied  to  extend  or  compress  the  spring,  as  if 


HELICAL  SPRINGS  109 

applied  directly  to  twist  a  piece  of  straight  wire  of  the  same 
material  with  a  lever 


This  may  be  easily  shown  by  a  model. 

The  safe  working  load  may  be  found  by  solving  for  Pf  in  (b) 
and  remembering  that  P  =  Pf 

(     . 


2.55Z)     2.55£ 

when  S  is  the  safe  shearing  strength. 

Substituting  this  value  of  P  in   (58)   we  have  for  the  safe 
deflection: 


45.  Square  Wire.  —  The  value  of  the  stress  for  a  square  section  is  : 

c,     4.24F 
~dT 
where  d  is  the  side  of  square. 

The  distortion  at  the  corners  caused  by  twisting  through  an 
angle  0  is: 

6d 


s  = 


IV2 


Equation  (c)  then  becomes: 


WDl 


The  three  principal  equations  (58)  ,  (59)  and  (60)  then  reduce  to: 

1.5PLD2 
~ 


The  square  section  is  not  so  economical  of  material  as  the 
round. 

46.  Experiments.  —  Tests  made  on  about  1700  tempered  steel 
springs  at  the  French  Spring  Works  in  Pittsburg  were  reported 
in  1901  by  Mr.  R.  A.  French.1  These  were  all  compression  springs 

1  Trans.  A.  S.  M.  E.,  Vol.  XXIII. 


110 


MACHINE  DESIGN 


of  round  steel  and  were  given  a  permanent  set  before  testing  by 
being  closed  coil  to  coil  several  times.  Table  XXXIII  gives 
results  of  these  experiments. 


(•BUOIS.10J, 
JO  lU810iy<K>Q 


•Snuds 
osoio  0}  ptK>T 


•pasoio 


-  H 


30 


u«ai\[ 


30 


CO  rH  rt  «  (M  (M  r-l  »  i-l  St  Ot  J-t         7-H  T-H  r-c  ^<  ^1  i-l  t-  «  T-I  TH  0»  T-H  rH  « 


OTtMecccinoti-^'eomeceocceo-'i'cococo! 


•dnoj£) 


HELICAL  SPRINGS  111 

The  apparent  variation  of  G  in  the  experiments 'is  probably 
due  to  differences  in  the  quality  of  steel  and  to  the  fact  that  the 
formula  for  G  in  the  case  of  helical  springs  is  an  approximate  one. 

The  same  may  be  said  of  the  values  of  S,  but  if  these  values  are 
used  in  designing  similar  springs  one  error  will  off-set  the  other. 

In  some  few  cases,  as  in  No.  18,  it  was  necessary  to  use  an 
abnormally  high  value  of  S  to  meet  the  conditions.  This  neces- 
sitated a  special  grade  of  steel,  and  great  care  in  manufacture. 
Such  a  spring  is  not  safe  when  subjected  to  sudden  and  heavy 
loads,  or  to  rapid  vibrations,  as  it  would  soon  break  under  such 
treatment;  if  merely  subjected  to  normal  stress,  it  would  last  for 
years. 

Springs  of  a  small  diameter  may  safely  be  subjected  to  a  higher 
stress  than  those  of  a  larger  diameter,  the  size  of  bar  being  the 
same.  The  safe  variation  of  S  with  R  cannot  yet  be  stated. 

There  is  an  important  limit  which  should  be  here  mentioned. 
Springs  having  two  small  a  diameter  as  compared  with  size  of  bar 
are  subjected  to  so  much  internal  stress  in  coiling  as  to  weaken 
the  steel.  A  spring,  to  give  good  service,  should  never  have  R  less 
than  3. 

The  size  of  bar  has  much  to  do  with  the  safe  value  of  S;  the 
probable  explanation  is  this:  A  large  bar  has  to  be  heated  to  a 
higher  temperature  in  working  it,  and  in  high  carbon  steel  this 
may  cause  deterioration;  when  tempered,  the  bath  does  not  affect 
it  so  uniformly,  as  may  be  seen  by  examining  the  fracture  of  a 
large  bar. 

The  above  facts  must  always  be  taken  into  consideration  in 
designing  a  spring,  whatever  the  grade  of  steel  used.  A  safe 
value  of  S  can  be  determined  only  by  one  having  an  accurate 
knowledge  of  the  physical  characteristics  of  the  steel,  the  pro- 
portions of  the  spring,  and  the  conditions  of  use. 

For  a  good  grade  of  steel  the  values  of  S  on  p.  112  have  been 
found  safe  under  ordinary  conditions  of  service,  the  value  of  G 
being  taken  as  14,500,000. 

For  bars  over  1|  in.  in  diameter  a  stress  of  more  than  100,000 
should  not  be  used.  Where  a  spring  is  subjected  to  sudden 
shocks  a  smaller  value  of  S  is  necessary. 

As  has  been  noted,  the  springs  referred  to  in  this  paper  were 
all  compression  springs.  Experience  has  shown  that  in  close 


112  MACHINE  DESIGN 

coil  or  extension  springs  the  value  of  G  is  the  same,  but  that  the 
safe  value  of  S  is  only  about  two-thirds  that  for  a  compression 
spring  of  the  same  dimensions. 

VALUES  OF  S 


... 

— 

d  — 

in.  or  less  
i   in  \jQ  3  in 

112,000 
110,000 

85,000 
80,000 

f  in.  to  \\  in  

105,000 

75,000 

47.  Spring  in  Torsion. — If  a  helical  spring  is  used  to  resist 
torsion  instead  of  tension  or  compression,  the  wire  itself  is 
subjected  to  a  bending  moment.  We  will  use  the  same  notation 
as  in  the  last  article,  only  that  P  will  be  taken  as  a  force  acting 

tangentially  to  the  circumference  of  the  spring  at  a  distance  -~- 

from  the  axis,  and  S  will  now  be  the  safe  transverse  strength  of 
the  wire,  having  the  following  values: 
$  =  60,000  for  spring  brass  wire 

=  90,000  to  125,000  for  cast  steel  tempered 
#  =  9,000,000  for  spring  brass  wire 

=  30,000,000  for  cast  steel  tempered. 
Let  6  =  angle  through  which  the  spring  is  turned  by  P. 
The  bending  moment  on  the  wire  will  be  the  same  throughout 

PD 
and  =  -  This  is  best  illustrated  by  a  model. 

£t 

To  entirely  straighten  the  wire  by  unwinding  the  spring  would 
require  the  same  force  as  to  bend  straight  wire  to  the  curvature 
of  the  helix. 

To  simplify  the  equations  we  will  disregard  the  obliquity  of 
the  helix,  then  will  l  =  nDn  and  the  radius  of  curvature 

D 

~  2" 

Let  M  =  bending  moment  caused  by  entirely  straightening 
the  wire;  then  by  mechanics 

.     El     2EI 


HELICAL  SPRINGS  113 

and  the  corresponding   angle  through   which  spriifg  is  turned 
is  2/m. 

But  it  is  assumed  that  a  force  P  with  a  radius  -=  turns  the 

£t 

spring  through  an  angle  6. 

PD     2EI      -0 

X 


2         D 

_  EIO  _EId 

Solving  for  0: 

'      PDl 

V=~<yWr  W 

Ztil 

and  if  wire  is  round 

-     10-2PM.  (64) 


The  bending  moment  for  round  wire  will  be 
PD     Sd* 
"102 

and  this  will  also  be  the  safe  twisting  moment  that  can  be 
applied  to  the  spring  when  S  =  working  strength  of  wire.  The 
safe  angle  of  deflection  is  found  by  substituting  this  value  of 


97  ^f 

Reducing:  0=||.  (66) 

48.  Flat  Springs. — Ordinary  flat  springs  of  uniform  rectangular 
cross-section  can  be  treated  as  beams  and  their  strength  and 
deflection  calculated  by  the  usual  formulas. 

In  such  a  spring  the  bending  and  the  stress  are  greatest  at 
some  one  point  and  the  curvature  is  not  uniform. 

To  correct  this  fault  the  spring  is  made  of  a  constant  depth 
but  varying  width. 

If  the  spring  is  fixed  at  one  end  and  loaded  at  the  other  the 
plan  should  be  a  triangle  with  the  apex  at  loaded  end.  If  it  is 
supported  at  the  two  ends  and  loaded  at  the  center,  the  plan 
should  be  two  triangles  with  their  bases  together  under  the  load 
forming  a  rhombus.  The  deflection  of  such  a  spring  is  one  and 
a  half  times  that  of  a  rectangular  spring. 


114  MACHINE  DESIGN 

As  such  a  spring  might  be  of  an  inconvenient  width,  a  com- 
pound or  leaf-spring  is  made  by  cutting  the  triangular  spring 
into  strips  parallel  to  the  axis,  and  piling  one  above  another  as 
in  Fig.  37. 

This  arrangement  does  not  change  the  principle,  save  that  the 
friction  between  the  leaves  may  increase  the  resistance  somewhat. 


FIG.  37. 

Let J,  =  length    of   span 
b  =  breadth  of  leaves 
t  =  thickness  of  leaves 
n  =  number  of  leaves 
TF  =  load   at   center 
A  =  deflection  at  center. 

S  and  E  may  be  taken  as  80,000  and  30,000,000  respectively. 
Strength: 

Wl     Snbt2 


(67) 

O  I 

Elasticity: 


Wl3  nbt3 


49.  Elliptic  and  Semi-elliptic  Springs.  —  Springs  as  they  are  usu- 
ally designed  for  service  differ  in  some  respects  from  those  just 
described,  as  may  be  seen  by  reference  to  Fig.  38.  A  band  is  used 


ELLIPTIC  SPRINGS 


115 


at  the  center  to  confine  the  leaves  in  place.  As  this  band  con- 
strains the  spring  at  the  center  it  is  best  to  consider  the  latter  as 

made  up  of  two  cantilevers  each  having  a  length  of  —     -  where  w 

2i 

is  the  width  of  band.  The  spring  usually  contains  several  full- 
length  leaves  with  blunt  ends,  the  remaining  leaves  being 
graduated  as  to  length  and  pointed  as  in  Fig.  38.  The  blunt 
full-length  leaves  constitute  a  cantilever  of  uniform  cross-section, 
while  the  graduated  leaves  form  a  cantilever  of  uniform  strength. 
Under  similar  conditions  as  to  load  and  fiber  stress  the  latter 
will  have  a  deflection  50  per  cent  greater  than  the  former.  Sup- 


FIG.  38. 

posing  that  there  is  no  initial  stress  between  the  leaves  caused 
by  the  band,  both  sets  must  have  the  same  deflection.  This 
means  that  more  than  its  proportion  of  the  load  will  be  carried 
by  the  full-length  set  and  consequently  it  will  have  a  greater 
fiber  stress.  This  difficulty  can  be  obviated  by  having  an  initial 
gap  between  the  graduated  set  and  the  full-length  set  and 
closing  this  with  the  band. 

If  this  gap  is  made  half  the  working  deflection  of  the  spring, 
the  total  deflection  of  the  graduated  set  under  the  working  load 
will  be  50  per  cent  greater  than  that  of  the  full-length  set  and  the 
fiber  stress  will  be  uniform. 

The  load  will  then  be  divided  between  the  two  sets  in  propor- 
tion to  the  number  of  leaves  in  each. 

One  of  the  full-length  leaves  must  be  counted  as  a  part  of  the 
graduated  set.  When  the  gap  is  closed  by  a  band  there  will  be 
an  initial  pull  on  the  band  due  to  the  deflection  of  the  spring. 

This  can  be  determined  for  any  given  spring  by  regarding  the 
two  sets  of  leaves  as  simple  beams  the  sum  of  whose  deflections 
under  the  pull  P  is  equal  to  the  depth  of  the  gap. 

Full  elliptic  springs  can  be  designed  in  a  similar  manner  but  the 
total  deflection  will  be  double  that  of  the  semi-elliptic  spring. 


116  MACHINE  DESIGN 

The  mathematical  discussion  of  which  the  following  is  an 
abstract  was  given  by  Mr.  E.  R.  Morrison,1  who,  as  far  as  the 
author  knows,  was  the  first  to  treat  the  subject  in  this  way. 

Let  c       =  whole  length  of  spring 
w      =  width  of  band 

I       =—  —  =  length  of  each  cantilever 
2i 

b  =  breadth  of  leaves 

t  =  thickness  of  one  leaf 

n  =  total  number  of  leaves 

n'  =  number  of  full-length  leaves 

n"  =  number  of  graduated  leaves 

..    n' 

r      —  ratio  — 
n 

S  =  maximum  fiber  stress  in  spring 

S'  =  maximum  fiber  stress  in  full-length  leaves 

A  =  total  deflection  of  spring 

A  '  =  total  deflection  of  full-length  leaves   if   unbanded 

A  "  =  total  deflection  of  graduated  leaves  if  unbanded 

P  =  total  load  on  spring 

P'  =  portion  of  load  on  one  end  of  full-length  leaves 

P"  —  portion  of  load  on  one  end  of  graduated  leaves. 

Assuming  that  the  maximum  stress  should  be  the  same  in  both 
parts: 

8'=  8" 


and 

^—^  (  \ 

P"~n"' 

The  deflections,  as  already  stated  on  preceding  page,  will  be 
unequal.     For  a  cantilever  of  uniform  section  (full-length  leaves)  : 


and  for  one  of  uniform  strength  (graduated  leaves)  : 

a  P//73 

A"=-w 

1  Mchy.,  N.  Y.,  Jan.,  1910. 


ELLIPTIC  SPRINGS  117 

But  from  (a) 


and 


But  also 


pn 

and  A"-A' 


Equation  (d)  gives  the  excess  of  the  deflection  of  the  graduated 
portion  over  that  of  the  full-length  portion  and  is  the  proper 
depth  of  gap  between  the  two  portions  before  banding.  To 
find  the  effect  of  the  banding: 

Let       Pb=  force  exerted  by  band 

d'  =  deflection  caused  by  band  in  full-length  leaves 
d"  =  deflection  caused  by  band  in  graduated  leaves. 
Then, 


and 

d"  = 


(Since  force  at  end  of  each  cantilever  =  - 

£t 


By  division  and  cancellation, 

2n"d 


Pi3 

The  depth  of  gap  =  <f  +<f'  =  ,-,  ,  3  from  equation  (d). 

Jit  not 

Combining: 


Enbt* 

m?  (h) 


118  MACHINE  DESIGN 

Equating  (f)  and  (h)  : 

PI 


/         n       \ 

En"bt*     \3n'  +  2n")  Enbt* 
Solving  for  P&  : 

Pb-        *'*"         P 

n(3n'+2n") 

Or  letting  n'  =  rn  n"  =  (l-r)  n 


and  this  equation  gives  the  force  exerted  by  the  band  in  terms 
of  the  total  load. 

The  woiking  deflection  of  the  spring  may  be  obtained  as 
follows: 

The  total  deflection  of  the  graduated  leaves  undei  the  load  P 
is  by  equation  (c)  : 

=  6P"l3  =  3PZ3 
~  En"bt3~  Enbt* 

But  a  part  of  this  total  is  produced  by  the  banding,  equation  (h)  : 

#>-(      3n'      X      PI* 
W  +  2n7V   Enbt3 

The  remaining  deflection  or  that  due  to  the  application  of  the 
load  P  is: 


But 


3ri  +  2n"I    Enbt3 
6          Pl3 


where  P'+P^^load    at   each  end  of  spring  and   (P'+P")Z  = 
bending  moment  at  band. 
Substituting  in  (j)  and  reducing: 

2        SZ2  ,PQ. 

-  (69) 


If  all  the  leaves  are  full  length  : 


ELLIPTIC  SPRINGS  119 

If  all  the  leaves  are  graduated: 

r  =  0  and  A  =  -=-• 

Hit 

PROBLEMS 

1.  A  spring  balance  is  to  weigh  50  Ib.  with  an  extension  of  2  in.,  the 
diameter  of  spring  being  f  in.  and  the  material,  tempered  steel. 

Determine  the  diameter  and  length  of  wire,  and  number  of  coils. 

2.  Determine  the  safe  twisting  moment  and  angle  of  torsion  for  the  spring 
in  example  1,  if  used  for  a  torsional  spring. 

3.  Test  values  of  G  and  S  from  data  given  in  Table  XXXIII. 

4.  By  using  above  table  design  a  spring  8  in.  long  to  carry  a  load  of  2  tons 
without  closing  the  coils  more  than  half  way. 

5.  Design  a  compound  flat  spring  for  a  locomotive  to  sustain  a  load  of 
16,000  Ib.  at  the  center,  the  span  being  40  in.,  the  number  of  leaves  12  and 
the  material  steel. 

6.  Determine  the  maximum  deflection  of  the  above  spring,  under  the 
working  load. 

7.  A  semi-elliptic  spring  has  9  leaves  in  all  and  6  graduated  leaves,  and 
the  load  on  each  end  is  P=4000  Ib.     Develop  formulas  for  the  fiber  stress 
in  each  set  of  leaves  if  there  is  no  initial  stress.     Determine  proper  breadth 
and  thickness  of  leaves  if  length  of  span  is  42  in. 

8.  In  Prob.  7  develop  a  formula  for  the  necessary  gap  to  equalize  the 
fiber  stresses. 

9.  In  Prob.  8  determine  the  pull  on  the  band  due  to  the  initial  stress. 

10.  A  semi-elliptic  spring  has  4  leaves  36  in.  long,  and  12   graduated 
leaves.     The  leaves  are  all  4  in.  wide  and  f  in.  thick,  and  the  band  at  the 
center  is  4  in.  wide.     If  there  is  no  initial  stress  find  the  share  of  the  load 
and  the  fiber  stress  on  each  set  of  leaves  when  there  is  a  load  of  6  tons  on 
the  center.     Also  determine  deflection. 

11.  In  Prob.  10,  determine  the  amount  of  gap  needed  to  equalize  the 
stresses  in  the  two  sets  of  leaves,  and  the  pull  on  the  band  at  the  center. 
Determine  the  deflection  under  the  load. 

12.  Measure  various  indicator  springs  and  determine  value  of  G   from 
rating  of  springs. 

13.  Measure  various  brass  extension  springs,  calculate  safe  static   load 
and  safe  stretch. 

14.  Make  an  experiment  on  torsion  spring  to  determine  distortion  under 
a  given  load  and  calculate  value  of  E. 

REFERENCES 

Vibration  of  Springs.     Am.  Mach.,  May  11,  1905. 
Tables  of  Loads  and  Deflections.     Am.  Mach.,  Dec.  20,  1906. 
The  Constructor.     Reuleaux. 


CHAPTER  VI 
SLIDING  BEARINGS 

50.  Slides  in  General. — The  surfaces  of  all  slides  should  have 
sufficient  area  to  limit  the  intensity  of  piessure  and  prevent 
forcing  out  ol  the  lubricant.     No  general  rule  can  be  given  for 
the  limit  of  pressure.     Tool  marks  parallel  to  the  sliding  motion 
should  not  be  allowed,  as  they  tend  to  start  grooving.     The 
sliding  piece  should  be  as  long  as  practicable  to  avoid  local  wear 
on  stationary  piece  and  for  the  same  reason  should  have  sufficient 
stiffness  to  prevent  springing.     A  slide  which  is  in  continuous 
motion  should  lap  over  the  guides  at  the  ends  of  stroke,  to  prevent 
the  wearing  of  shoulders  on  the  latter  and  the  finished  surfaces 
of  all  slides  should  have  exactly  the  same  width  as  the  surfaces 
on  which  they  move  for  a  similar  reason. 

Where  there  are  two  parallel  guides  to  motion  as  in  a  lathe  or 
planer  it  is  better  to  have  but  one  of  these  depended  upon  as  an 
accurate  guide  and  to  use  the  other  merely  as  a  support.  It 
must  be  remembered  that  any  sliding  bearing  is  but  a  copy  of 
the  ways  of  the  machine  on  which.it  was  planed  or  ground  and  in 
turn  may  reproduce  these  same  errors  in  other  machines.  The 
interposition  of  h  and -sci  aping  is  the  only  cure  for  these  hereditary 
complaints. 

In  designing  a  slide  one  must  consider  whether  it  is  accuracy 
of  motion  that  is  sought,  as  in  the  ways  of  a  planer  or  lathe,  or 
accuracy  of  position  as  in  the  head  of  a  milling  machine.  Slides 
may  be  divided  according  to  their  shapes  into  angular,  flat  and 
circular  slides. 

51.  Angular  Slides. — An  angular  slide  is  one  in  which  the 
guiding  surface  is  not  normal  to  the  direction  of  pressure.     There 
is  a  tendency  to  displacement  sideways,  which  necessitates  a 
second    guiding    surface    inclined    to    the    first.     This    oblique 
pressure  constitutes  the  principal  disadvantage  of  angular  slides. 

120 


ANGULAR  SLIDES  121 

Their  principal  advantage  is  the  fact  that  they  are 'either  self- 
adjusting  for  wear,  as  in  the  ways  of  lathes  and  planers,  or 
require  at  most  but  one  adjustment. 

Fig.  39  shows  one  of  the  V's  of  an  ordinary  planing  machine. 
The  platen  is  held  in  place  by  gravity.  The  angle  between  the 
two  surfaces  is  usually  90  degrees  but  may  be  more  in  heavy 
machines.  The  gro.oves  g,  g  are  intended  to  hold  the  oil  in  place; 
oiling  is  sometimes  effected  by  small  rolls  recessed  into  the  lower 
piece  and  held  against  the  platen  by  springs. 

The  principal  advantage  of  this  form  of  way  is  its  ability  to 
hold  oil  and  the  great  disadvantage,  its  faculty  for  catching  chips 
and  dirt. 

Fig.  40  shows  an  inverted  V  such  as  is  common  on  the  ways  of 
engine  lathes.  The  angle  is  about  the  same  as  in  the  preceding 
form  but  the  top  of  the  V  should  be  rounded  as  a  precaution 
against  nicks  and  bruises. 


FIG.  39.  FIG.  40. 

The  inverted  V  is  preferred  for  lathes  since  it  will  not  catch 
dirt  and  chips.  It  needs  frequent  lubrication  as  the  oil  runs  off 
rapidly.  Some  lathe  carriages  are  provided  with  extensions 
filled  with  oily  felt  or  waste  to  protect  the  ways  from  dirt  and 
keep  them  wiped  and  oiled.  Side  pressure  tends  to  throw  the 
carriage  from  the  ways;  this  action  may  be  prevented  by  a  heavy 
weight  hung  on  the  carriage  or  by  gibbing  the  carriage  at  the 
back  (see  Fig.  46).  The  objection  to  this  latter  form  of  con- 
struction is  the  fact  that  it  is  practically  impossible  to  make 
and  keep  the  two  V's  and  the  gibbed  slide  all  parallel. 

Fig.  41  shows  a  compound  V  sometimes  used  on  heavy  ma- 
chines. The  obtuse  angle  (about  150  degrees)  takes  the  heavy 


122 


MACHINE  DESIGN 


vertical  pressure,  while  the  sides,  inclined  only  8  or  10  degrees, 
take  any  side  pressure  which  may  develop. 

52.  Gibbed  Slides. — All  slides  which  are  not  self-adjusting  for 
wear  must  be  provided  with  gibs  and  adjusting  screws.  Fig.  42 
shows  the  most  common  form  as  used  in  tool  slides  for  lathes  and 
planing  machines. 


g 


FIG.  41. 


FIG.  42. 


The  angle  employed  is  usually  60  degrees;  notice  that  the 
corners  c  c  are  clipped  for  strength  and  to  avoid  a  corner  bearing; 
notice  also  the  shape  of  gib.  It  is  better  to  have  the  points  of 
screws  coned  to  fit  gib  and  not  to  have  flat  points  fitting  recesses 
in  gib.  The  latter  form  tends  to  spread  the  joint  apart  by 

forcing  the  gib  down.  If  the  gib  is 
too  thin  it  will  spring  under  the  screws 

^^          and  cause  uneven  wear. 

\~  f^5\  The  cast-iron  gib,  Fig.  43,  is  free 

from  this  latter  defect  but  makes 
the  slide  rather  clumsy.  The  screws, 
however,  are  more  accessible  in  this 
form.  Gibs  are  sometimes  made 

slightly  tapering  and  adjusted  by  a  screw  and  nut  giving  endwise 
motion. 


FIG.  43. 


53.  Flat  Slides. — This  type  of  slide  requires  adjustment  in 
two  directions  and  is  usually  provided  with  gibs  and  adjusting 
screws.  Flat  ways  on  machine  tools  are  the  rule  in  English 
practice  and  are  gradually  coming  into  use  in  this  country. 
Although  more  expensive  at  first  and  not  so  simple  they  are 
more  durable  and  usually  more  accurate  than  the  angular  ways. 

Fig.  44  illustrates  a  flat  way  for  a  planing  machine.     The  other 


FLAT  SLIDES 


123 


way  would  be  similar  to  this  but  without  adjustment.  The 
normal  pressure  and  the  friction  are  less  than  with  angular  ways 
and  no  amount  of  side  pressure  will  lift  the  platen  from  its 
position. 


FIG.  44. 


FIG.  45. 


Fig.  45  shows  a  portion  of  the  ram  of  a  shaping  machine  and 
illustrates  the  use  of  an  L  gib  for  adjustment  in  two  directions. 
Fig.  46  shows  a  gibbed  slide  for  holding  down  the  back  of  a  lathe 
carriage  with  two  adjustments. 


LnJ 


FIG.  46. 


FIG.  47. 


The  gib  g  is  tapered  and  adjusted  by  a  screw  and  nuts.  The 
saddle  of  a  planing  machine  or  the  table  of  a  shaper  usually  has 
a  rectangular  gibbed  slide  above  and  a  taper  slide  below,  this 
form  of  the  upper  slide  being  necessary  to  hold  the  weight  of  the 


124 


MACHINE  DESIGN 


overhanging  metal  (see  Fig.  47).  Some  lathes  and  planers  are 
built  with  one  V  or  angular  way  for  guiding  the  carriage  or 
platen  and  one  flat  way  acting  merely  as  a  support. 


B 


54.  Circular  Guides. — Examples  of  this  form  may  be  found  in 

the  column  of  the  drill  press 
and  the  overhanging  arm  of 
the  milling  machine.  The 
cross  heads  of  steam  engines 
are  sometimes  fitted  with 
circular  guides;  they  are  more 
frequently  flat  or  angular. 
One  advantage  of  the  circular 
form  is  the  fact  that  the  cross 


FIG   48<  head  can  adjust  itself  to  bring 

the  wrist  pin  parallel  to  the 

crank   pin.     The  guides  can  be  bored  at  the  same  setting  as 
the  cylinder  in  small  engines  and  thus  secure  good  alignment. 
Fig.  48  illustrates  various  shapes  of  cross  head  slides  in  common 
use. 

55.  Stuffing  Boxes. — In  steam  engines  and  pumps  the  glands 
for  holding  the  steam  and  water  packing  are  the  sliding  bearings 


FIG.  49.' 

which  cause  the  greatest  friction  and  the  most  trouble.  Fig.  49 
shows  the  general  arrangement.  B  is  the  stuffing  box  attached 
to  the  cylinder  head;  R  is  the  piston  rod;  G  the  gland  adjusted  by 


STUFFING  BOX  125 

nuts  on  the  studs  shown;  P  the  packing  contained  in, a  recess  in 
the  box  and  consisting  of  rings,  either  of  some  elastic  fibrous 
material  like  hemp  and  woven  rubber  cloth  or  of  some  soft  metal 
like  Babbitt  metal.  The  pressure  between  the  packing  and  the 
rod,  necessary  to  prevent  leakage  of  steam  or  water,  is  the  cause 
of  considerable  friction  and  lost  work.  Experiments  made  from 
time  to  time  in  the  laboiatories  of  the  Case  School  have  shown 
the  extent  and  manner  of  variation  of  this  friction.  The  results 
for  steam  packings  may  be  summarized  as  follows: 

1.  That  the  softer  rubber  and  graphite  packings,  which  are  self- 
adjusting  and  self -lubricating,  as  in  Nos.  2,  3,  77  8,  and  11,  con- 
sume less  power  than  the  harder  varieties.     No.   17,  the  old 
braided    flax    style,     gives    very    good    results.     (See   Table 
(XXXIV.) 

2.  That  oiling  the  rod  will  reduce  the  friction  with  any  packing, 

3.  That  there  is  almost  no  limit  to  the  loss  caused  by  the 
injudicious  use  of  the  monkey-wrench. 

4.  That  the  power  loss  varies  almost  directly  with  the  steam 
pressure  in  the  harder  varieties,  while  it  is  approximately  con- 
stant with  the  softer  kinds. 

The  diameter  of  rod  used — 2  in. — would  be  appropriate  for 
engines  from  50  to  100  horse-power.  The  piston  speed  was  about 
140  ft.  per  minute  in  the  expeiiments,  and  the  horse-power  varied 
from  .036  to  .400  at  50  Ib.  steam  pressure,  with  a  safe  average 
for  the  softer  class  of  packings  of  .07  horse-power. 

At  a  piston  speed  of  600  ft.  per  minute,  the  same  friction  would 
give  a  loss  of  from  .154  to  1.71  with  a  working  average  of  .30 
horse-power,  at  a  mean  steam  pressure  of  50  Ib. 

In  Table  XXXIV  Nos.  6,  14,  15,  and  16  are  square,  hard  rubber 
packings  without  lubricants. 

Similar  experiments  on  hydraulic  packings  under  a  water 
pressure  varying  from  10  to  80  Ib.  per  square  inch  gave  results 
as  shown  in  Table  XXXVI. 

The  figures  given  are  for  a  2-in.  rod  running  at  an  average 
piston  speed  of  140  ft.  per  minute. 


126 


MACHINE  DESIGN 


TABLE  XXXIV 


Average 

Horse- 

Total 

horse- 

power 

Kind  of 
packing 

No 
trials 

time  of 
run  in 

power 
con- 
sumed 

con- 
sumed 
at  50 

Remarks  on  leakage,  etc. 

minutes 

by  each 

Ib.  pres- 

box 

sure 

1 

5 

22 

.091 

.085 

Moderate  leakage. 

2 

8 

40 

.049 

.048 

Easily  adjusted;  slight  leakage. 

3 

5 

25 

.037 

.036 

Considerable  leakage. 

4 

5 

25 

.159 

.176 

Leaked  badly. 

5 

5 

25 

.095 

.081 

Oiling  necessary;  leaked  badly. 

6 

5 

25 

.368 

.400 

Moderate  leakage. 

7 

5 

25 

.067 

.067 

Easily  adjusted  and  no  leakage. 

8 

5 

25 

.082 

.082 

Very  satisfactory;  slight  leakage. 

9 

3 

15 

.200 

.182 

Moderate  leakage. 

10 

3 

.275 

Excessive  leakage. 

11 

5 

25 

.157 

.172 

Moderate  leakage. 

12 

5 

25 

.266 

.330 

Moderate  leakage. 

13 

5 

25 

.162 

.230 

No  leakage;  oiling  necessary. 

14 

5 

25 

.176 

.276 

Moderate  leakage;  oiling  necessary 

15 

5 

25 

.233 

.255 

Difficult  to  adjust;  no  leakage. 

16 

5 

25 

.292 

.210 

Oiling  necessary;  no  leakage. 

17 

5 

25 

.128 

.084 

No  leakage. 

TABLE  XXXV 


Kind  of 
packing 

Horse-power  consumed  by  each  box,  when  pressure  was 
applied  to  gland  nuts  by  a  7-in.  wrench 

Horse-power 
before  and  after 
oiling  rod 

51b. 

81b. 

10  Ib. 

12  Ib. 

14  Ib. 

16  Ib. 

Dry 

Oiled 

1 
3 
4 
5 
6 
7 
8 
9 
11 
12 
13 
15 
16 
17 

.120 

.136 

.021 
.123 

.055 
.154 

.248 
.220 
.348 
.126 
.363 
.666 

.430 
.228 
.500 

.303 

.260 
.535 

.330 
.520 

.390 

.340 
.533 

.323 
.067 
.533 
.666 
.454 
.454 

.194 
.053 
.236 
.636 
.176 
.122 

.405 
.161 
.317 
526 

.454 
.242 
.394 

.359 
.582 

.454 



.327 
.198 

.860 

.277 

.380 

!  

STUFFING  BOXES 
TABLE  XXXVI 


127 


No.  of 
packing 

Av.  H.  P. 

at  20  Ib. 

Av.  H.  P. 

at  70  Ib. 

Max. 
H.  P. 

Min. 
H.  P. 

Av.  H.  P. 

for  entire 
test 

1 

.077 

.351 

.452 

.024 

.259 

2 

.422 

.500 

.512 

.167 

.410 

3 

.130 

.178 

.276 

.035 

.120 

4 

.184 

.195 

.230 

.142 

.188 

5 

.146 

.162 

.285 

.069 

.158 

6 

.240 

.200 

.255 

.071 

.186 

7 

.127 

.192 

.213 

.095 

.154 

8 

.153 

.174 

.238 

.112 

.165 

9 

.287 

.469 

.535 

.159 

.389 

10 

.151 

.160 

.226 

.035 

.103 

11 

.141 

.156 

.380 

.064 

.177 

12 

.053 

.095 

.143 

.035 

.090 

Packings  Nos.  5,  6,  10  and  12  are  braided  flax  with  graphite 
lubrication  and  are  best  adapted  for  low  pressures.  Packings 
Nos.  3,  4  and  7  are  similar  to  the  above  but  have  paraffine  lubri- 
cation. Packings  Nos.  2  and  9  are  square  duck  without  lubri- 
cant and  are  only  suitable  for  very  high  pressures,  the  friction 
loss  being  approximately  constant. 


PROBLEMS 

Make  a  careful  study  and  sketch  of  the  i 
following  machines  and  analyze  as  to  (a) 
Adjustment,  (d)  Lubrication. 

1.  An  engine  lathe. 

2.  A  planing  machine. 

3.  A  shaping  machine. 

4.  A  milling  machine. 

5.  An  upright  drill. 

6.  A  Corliss  engine. 

7.  A  locomotive  engine. 

8.  A  gas-engine. 

9.  An  air-compressor. 


liding  bearings  on  each  of  the 
Purpose.     (6)  Character,     (c) 


CHAPTER  VII 
JOURNALS,  PIVOTS  AND  BEARINGS 

56.  Journals. — A  journal  is  that  part  of  a  rotating  shaft  which 
rests  in  the  bearings  and  is  of  necessity  a  surface  of  revolution, 
usually  cylindrical  or  conical.  The  material  of  the  journal  is 
generally  steel,  sometimes  soft  and  sometimes  hardened  and 
ground. 

The  material  of  the  bearing  should  be  softer  than  the  journal 
and  of  such  a  quality  as  to  hold  oil  readily.  The  cast  metals 
such  as  cast  iron,  bronze  and  Babbitt  metal  are  suitable  on 
account  of  their  poious,  granular  character.  Wood,  having  the 
grain  normal  to  the  bearing  surface,  is  used  where  water  is  the 
lubiicant,  as  in  water  wheel  steps  and  stern  bearings  of  propellers. 

Bearing  materials  may  naturally  be  divided  into  soft  and  hard 
metals.  The  standard  soft  metal  is  so-called  "  genuine  Babbitt," 
of  the  following  composition: 


Tin,  85  to  89  per  cent. 
Copper,  2  to  5  per  cent. 
Antimony,  7  to  10  pei  cent. 


The  substitution  of  lead  for  tin  and  the  omission  of  the  copper 
makes  a  cheaper  and  softer  metal  suitable  for  low  pressures  and 
speeds.  The  addition  of  more  antimony  hardens  the  metal. 

The  hard  metals  include  the  various  brasses  and  bronzes 
tanging  from  soft  yellow  brass  to  phosphor  and  aluminum 
bronzes. 

Professor  R.  C.  Carpenter  recommends  as  suitable  for  a  bearing 
an  aluminum  bronze  whose  composition  is: 

Aluminum,  50  per  cent. 
Zinc,  25  per  cent. 
Tin,  25  per  cent. 

This  metal  is  light,  fairly  hard,  and  will  not  melt  readily.1 

1  Trans.  A.  S.  M.  E.,  Vol.  XXVII,  p.  425. 

128 


JOURNAL  BEARINGS 


129 


57.  Adjustment. — Bearings  wear  more  or  less  lapidly  with  use 
and  need  to  be  adjusted  to  compensate  for  the  wear.  The 
adjustment  must  be  of  such  a  character  and  in  such  a  direction 
as  to  take  up  the  wear  and  at  the  same  time  maintain  as  far  as 
possible  the  correct  shape  of  the  bearing.  The  adjustment 
should  then  be  in  the  line  of  the  greatest  pressure. 

Fig.  50  illustrates  some  of  the  more  common  ways  of  adjusting 
a  bearing,  the  arrows  showing  the  direction  of  adjustment  and 
presumably  the  direction  of  pressure;  (a)  is  the  most  usual  where 
the  principal  wear  is  vertical,  (d)  is  a  form  frequently  used  on 
the  main  journals  of  engines  when  the  wear  is  in  two  directions, 


FIG.  50. 


FIG.  51. 


horizontal  on  account  of  the  steam  pressure  and  vertical  on 
account  of  the  weight  of  shaft  and  fly-wheel.  All  of  these  are 
more  or  less  imperfect  since  the  bearing,  after  wear  and  adjust- 
ment, is  no  longer  cylindrical  but  is  made  up  of  two  or  more 
approximately  cylindrical  surfaces. 

A  bearing  slightly  conical  and  adjusted  endwise  as  it  wears,  is 
probably  the  closest  approximation  to  correct  practice. 

Fig.  51  shows  the  main  bearing  of  the  Porter-Allen  engine, 
one  of  the  best  examples  of  a  four  part  adjustment.  The  cap  is 
adjusted  in  the  normal  way  with  bolts  and  nuts;  the  bottom  can 
be  raised  and  lowered  by  liners  placed  underneath;  the  cheeks 
can  be  moved  in  or  out  by  means  of  the  wedges  shown.  Thus 
it  is  possible  not  only  to  adjust  the  bearing  for  wear  but  to  align 
the  shaft  perfectly. 

A  three  part  bearing  for  the  main  journal  of  an  engine  is 


130 


MACHINE  DESIGN 


shown  in  Fig.  52.     In  this  bearing  there  is  one  horizontal  adjust- 
ment instead  of  two  as  in  Fig.  51. 

The  main  bearing  of  the  spindle  in  a  lathe,  as  shown  in  Fig.  53, 
offers  a  good  example  of  symmetrical  adjustment.     The  head- 


FIG.  52. 


FIG.  53. 


stock  A  has  a  conical  hole  to  receive  the  bearing  B,  which  latter 
can  be  moved  lengthwise  by  the  nuts  FG.  The  bearing  may  be 
split  into  two,  three  or  four  segments  or  it  may  be  cut  as  shown 
in  e,  Fig.  50,  and  sprung  into  adjustment.  A  careful  distinction 


FIG.  54. 


must  be  made  between  this  class  of  bearing  and  that  before 
mentioned,  where  the  journal  itself  is  conical  and  adjusted  end- 
wise. A  good  example  of  the  latter  form  is  seen  in  the  spindles 
of  many  milling  machines. 

Fig.  54  shows  the  spindle  of  an  engine  lathe  complete  with  its 
two  bearings.     The  end  thrust  is  taken  by  a  fiber  washer  backed 


JOURNAL  BEARINGS 


131 


by  an  adjusting  collar  and  check  nut.     Both  bearings  belong  to 
the  class  shown  in  Fig.  53. 

A  conical  journal  with  end  adjustment  is  illustrated  in  Fig.  55, 
which  shows  the  spindle  of  a  milling  machine.  The  front  journal 
is  conical  and  is  adjusted  for  weai  by  drawing  it  back  into  its 
bearing  with  the  nut.  The  rear  journal  on  the  other  hand  is 
cylindrical  and  its  bearing  is  adjusted  as  are  those  just  described. 
The  end  thrust  is  taken  by  two  loose  rings  at  the  front  end  of  the 
spindle. 


FIG.  55. 

58.  Lubrication. — The  bearings  of  machines  which  run  inter- 
mittently, like  most  machine  tools,  are  oiled  by  means  of  simple 
oil  holes,  but  machinery  which  is  in  continuous  motion  as  is  the 
case  with  line  shafting  and  engines  requires  some  automatic 
system  of  lubrication.     There  is  not  space  in  this  book  for  a 
detailed  description  of  all  the  various  types  of  oiling  devices  and 
only  a  general  classification  will  be  attempted. 
Lubrication  is  effected  in  the  following  ways: 
1.   By  grease  cups. 
By  oil  cups. 

By  oily  pads  of  felt  or  waste. 
By  oil  wells  with  rings  or  chains  for  lifting  the  oil. 
By  centrifugal  force  through  a  hole  in  the  journal  itself. 
Grease  cups  have  little  to  recommend  them  except  as  auxiliary 
safety  devices.     Oil  cups  are  various  in  their  shapes  and  methods 
of  operation  and  constitute  the  cheap  class  of  lubricating  devices. 
They  may  be  divided  according  to  their  operation  into  wick  oilers, 
needle  feed,  and  sight  feed.     The  two  first  mentioned  are  nearly 
obsolete  and  the  sight-feed  oil  cup,  which  drops  the  oil  at  regular 


132 


MACHINE  DESIGN 


intervals  through  a  glass  tube  in  plain  sight,  is  in  common  use. 
The  best  sight-feed  oiler  is  that  which  can  be  readily  adjusted 
as  to  time  intervals,  which  can  be  turned  on  or  off  without  dis- 
turbing the  adjustment  and  which  shows  clearly  by  its  appear- 
ance whether  it  is  turned  on.  On  engines  and 
electric  machinery  which  are  in  continuous  use 
day  and  night,  it  is  very  important  that  the 
oiler  itself  should  be  stationary,  so  that  it 
may  be  rilled  without  stopping  the  machinery. 
A  modern  sight-feed  oiler  for  an  engine  is 
illustrated  in  Fig.  56.  T  is  the  glass  tube  where 
the  oil  drop  is  seen.  The  feed  is  regulated  by 
the  nut  Nt  while  the  lever  L  shuts  off  the 
oil.  Where  the  lever  is  as  shown  the  oil  is 
|  dropping,  when  horizontal  the  oil  is  shut  off. 

The  nut  can  be  adjusted  once  for  all,  and  the 
position  of  the  lever  shows  immediately  whether 
or  not  the  cup  is  in  use. 

In  modern  engines  particular  attention  has 
been  paid  to  the  problem  of  continuous  oiling. 
The   oil   cups  are   all   stationary  and   various 
ingenious  devices  are  used  for  catching  the  drops  of  oil  from 
the  cups  and   distributing  them  to  the  bearing  surfaces.     For 
continuous    oiling    of   stationary   bearings,   as   in  line  shafting 
and  electric  machinery,  an  oil  well  below  the  bearing  is  preferred, 
with   some   automatic  means  of  pumping 
the  oil  over  the  bearing,  when  it  runs  back 
by  gravity  into  the  well.     Porous  wicks  and 
pads  acting  by  capillary  attraction  are  un- 
certain in  their  action  and  liable  to  become 
clogged.      For   bearings   of    medium   size, 
one  or  more  light  steel  rings  running  loose 
on  the  shaft  and  dipping  into  the  oil,  as 
shown  in  Fig.  57,  are  the  best.     For  large 
bearings     flexible    chains    are    employed 
which  take  up  less  room  than  the  ring. 

Cases  have  been  reported,  however,  where  suction  oilers  on 
line  shafting  have  proved,  more  efficient  than  ring  oilers.  One 
instance  is  quoted  where  a  suction  oiler  has  been  in  continuous 


FIG.  56. 


FIG.  57. 


LUBRICATION 


133 


use  for  nearly  thirty  years  and  has  worked  perfectly  during 
that  entire  period.1  Much  depends  on  the  care  of  such  devices, 
the  prevalence  or  absence  of  dust,  and  the  quality  of  oil  used. 
Centrifugal  oilers  are  most  used  on  parts  which  cannot  readily 


r—\ 


FIG.  58. 

be  oiled  when  in  motion,  such  as  loose  pulleys  and  the  crank 
pins  of  engines. 

Fig.  58  shows  two  such  devices  as  applied  to  an  engine.  In 
A  the  oil  is  supplied  by  the  waste  from  the  main  journal;  in  B 
an  external  sight-feed  oil  cup  is  used  which  supplies  oil  to  the 
central  revolving  cup  C. 


FIG.  59. 

Loose  pulleys  or  pulleys  running  on  stationary  studs  are  best 
oiled  from  a  hole  running  along  the  axis  of  the  shaft  and  thence 
out  radially  to  the  surface  of  the  bearing.  See  Fig.  59.  A  loose 
bushing  of  some  soft  metal  perforated  with  holes  is  a  good  safety 
device  for  loose  pulleys. 

1  Trans.  A.  S.  M.  E.,  Vol.  XXVII,  p.  488. 


134  MACHINE  DESIGN 

Note:  For  adjustable  pedestal  and  hanging  bearings  see  the 
chapter  on  shafting. 

59.  Friction  of  Journals  : 

Let   TP  =  the  total  load  of  a  journal  in  pounds 
£=the  length  of  journal  in  inches 
d  =  ihe  diameter  of  journal  in  inches 
N  =  number  of  revolutions  per  minute 
v  =  velocity  of  rubbing  in  feet  per  minute 
F=friction  at  surface  of  journal  in  pounds 

=  W  tan  W  nearly,  where  W  is  the  angle  of  repose  for 
the  two  materials. 

If  a  journal  is  properly  fitted  in  its  bearing  and  does  not 
bind,  the  value  of  F  will  not  exceed  W  tan  W  and  may  be  slightly 
less.  The  value  of  tan  W  varies  according  to  the  materials  used 
and  the  kind  of  lubrication,  from  .05  to  .01  or  even  less.  See 
experiments  described  in  Art.  62.  The  work  absorbed  in  friction 
may  be  thus  expressed  : 

ndN    TidNWtanW  .,    ,,  .  /rrrv. 

<y-  =  --    --  ft.  Ibs.  per  mm.        (70) 


60.  Limits  of  Pressure.  —  Too  great  an  intensity  of  pressure 
between  the  surface  of  a  journal  and  its  bearing  will  force  out 
the  lubricant  and  cause  heating  and  possibly  "seizing."  The 
safe  limit  of  pressure  depends  on  the  kind  of  lubricant,  the 
manner  of  its  application  and  upon  whether  the  pressure  is  con- 
tinuous or  intermittent.  The  projected  area  of  a  journal,  or  the 
product  of  its  length  by  its  diameter,  is  used  as  a  divisor. 

The  journals  of  railway  cars  offer  a  good  example  of  con- 
tinuous pressure  and  severe  service.  A  limit  of  300  Ib.  per 
square  inch  of  projected  area  has  been  generally  adopted  in  such 
cases. 

In  the  crank  and  wrist  pins  of  engines,  the  reversal  of  pressure 
diminishes  the  chances  of  the  lubricant  being  squeezed  out,  and 
a  pressure  of  500  Ib.  per  square  inch  is  generally  allowed. 

The  use  of  heavy  oils  or  of  an  oil  bath,  and  the  employment 
of  harder  materials  for  the  journal  and  its  bearing  allow  of  even 
greater  pressures. 


BEARING  PRESSURES  135 

Professor  Barr's  investigations  of  steam  engine  proportions1 
show  that  the  pressure  per  square  inch  on  the  cross-head  pin 
varies  from  ten  to  twenty  times  that  on  the  piston,  while  the 
intensity  of  pressure  on  the  crank  pin  is  from  two  to  eight  times 
that  on  the  piston.  Allowing  a  mean  pressure  on  the  piston  of 
50  Ib.  per  square  inch  would  give  the  following  range  of  pressures: 

Minimum    Maximum 

Wrist  pins 500  1,000 

Crank  pins 100  400 

The  larger  values  for  the  wrist  pins  are  allowable  on  account 
of  the  comparatively  low  velocity  of  rubbing.  Naturally  the 
larger  values  for  the  pressure  are  found  in  the  low-speed  engines. 

A  discussion  of  the  subject  of  bearings  is  reported  in  the  trans- 
actions of  the  American  Society  of  Mechanical  Engineers  for 
1905-06  and  some  valuable  data  are  furnished. 

Mr.  George  M.  Basford  says  that  the  bearing  areas  of  locomo- 
tive journals  are  determined  chiefly  by  the  possibilities  of  lubri- 
cation. Crank  pins  may  be  loaded  to  from  1500  to  1700  Ib.  per 
square  inch,  since  the  reciprocation  of  the  rods  makes  lubrication 
easy.  Wrist  pins  have  been  loaded  as  high  as  4000  Ib.  per  square 
inch,  the  limited  arc  of  motion  and  the  alternating  pressures 
making  this  possible. 

Locomotive  driving  journals  on  the  other  hand  are  limited 
to  the  following  pressures: 

Passenger  locomotives 190  Ib.  per  square  inch. 

Freight  locomotives 200  Ib.  per  square  inch. 

Switching  locomotives 220  Ib.  per  square  inch. 

Cars  and  tender  bearings 300  Ib.  per  square  inch. 

Mr.  H.  G.  Reist  gives  some  figures  on  the  practice  of  the 
General  Electric  Company,  for  motors  and  generators. 

This  company  allows  from  30  to  80  Ib.  pressure  per  square 
inch  with  an  average  value  of  from  40  to  45  Ib.  The  rubbing 
speeds  vary  from  40  ft.  to  1200  ft.  per  minute.  Mr.  Reist  quotes 
approvingly  the  formula  of  Dr.  Thurston's,  viz.:  That  the 
product  of  the  pressure  in  pounds  per  square  inch  and  the 
rubbing  speed  in  feet  per  minute  should  not  exceed  50,000. 

The  ratio  of  length  to  diameter  of  journal  is  given  as  about 
3.1  but  a  smaller  ratio  is  used  in  special  cases. 

1  Trans.  A.  S.  M.  E.,  Vol.  XVIII. 


136  MACHINE  DESIGN 

Oil  rings  placed  not  further  than  8  in.  apart  have  given  good 
results  for  many  years.  For  bearings  more  than  1  ft.  in  diameter, 
a  forced  circulation  of  oil  is  recommended  to  carry  off  the  heat 
generated. 

The  practice  of  one  of  the  largest  firms  of  Corliss  engine  builders 
in  this  country  may  be  summarized  as  follows: 

All  bearings  are  lined  with  best  Babbitt  metal,  cast,  hammered 
in  place  and  bored. 

Lubrication  is  effected  by  pressure  oil  cups,  the  oil  dropping 
from  the  cup  into  a  cored  pocket  in  the  top  shell  of  the  bearing, 
this  pocket  being  filled  with  waste. 

The  speed  of  the  shafts  is  between  75  and  150  revolutions  per 
minute  and  the  allowable  pressure  on  the  journal  is  140  Ib. 
per  square  inch  of  projected  area.  (This  is  exclusive  of  steam 
pressure.) 

The  bearings  of  horizontal  engines  are  usually  four-part  shells 
with  the  cap  separate  from  the  upper  shell  and  the  lower  shell 
resting  on  a  rib  at  center,  which  makes  it  self -adjustable. 

The  bearings  of  vertical  engines  are  two-part  shells. 

A  careful  reading  of  the  whole  discussion  will  repay  any  one 
who  has  to  design  shaft  bearings  of  any  description.1 

61.  Heating  of  Journals. — The  proper  length  of  journals 
depends  on  the  liability  of  heating. 

The  energy  or  work  expended  in  overcoming  friction  is  con- 
verted into  heat  and  must  be  conveyed  away  by  the  material 
of  the  rubbing  surfaces.  If  the  ratio  of  this  energy  to  the  area 
of  the  surface  exceeds  a  certain  limit,  depending  on  circumstances, 
the  heat  will  not  be  conveyed  away  with  sufficient  rapidity  and 
the  bearing  wrill  heat. 

The  area  of  the  rubbing  surface  is  proportional  to  the  projected 
area  or  product  of  the  length  and  diameter  of  the  journal,  and 
it  is  this  latter  area  which  is  used  in  calculation. 

Adopting  the  same  notation  as  is  used  in  Art.  59,  we  have  from 
equation  (70). 

ndNWtanW    , 

the  .work  of  friction  = =— .  ft.  Ib.  per  mm. 

1 2i 

or  =  xdNWtan¥  inch  pounds. 
1  Trans.  A.  S.  M.  E.,  Vol.  XXVII. 


LENGTH  OF  JOURNALS  137 

The  work  per  square  inch  of  projected  area  is  then: 
ndNWtanW     TiNWtanW. 

w=      ~~W~  ~T~  (a) 

Solving  in  (a)  for  I 

l  =  - °^—'  (b) 

w 

Let  -    —jfr^C  a  coefficient  whose  value  is  to  be  obtained  by 
ntanW 

experiment;  then 

WN  WN 

C  =  -j-  andZ  =  -£--  (71) 

Crank  pins  of  steam  engines  have  perhaps  caused  more 
trouble  by  heating  than  any  other  form  of  journal.  A  com- 
parison of  eight  different  classes  of  propellers  in  the  old  U.  S.  Navy 
showed  an  average  value  for  C  of  350,000. 

A  similar  average  for  the  crank  pins  of  thirteen  screw  steamers 
in  the  French  Navy  gave  (7  =  400,000. 

Locomotive  crank  pins  which  are  in  rapid  motion  through  the 
cool  outside  air  allow  a  much  larger  value  of  C,  sometimes  more 
than  a  million. 

Examination  of  ten  modern  stationary  engines  shows  an 
average  value  of  C  =  200,000  and  an  average  pressure  per  square 
inch  of  projected  area  =  300  Ib. 

The  investigations  of  Professor  Barr  above  referred  to  show  a 
wide  variation  in  the  constants  for  the  length  of  crank  pins  in 


stationary  engines.     He  prefers  to  use  the  formula  :  I  =  K-^-  +  B 

Li 

where  K  and  B  are  constants  and  L  =  length  of  stroke  of  engine 
in  inches.     We  may  put  this  in  another  form  since: 

HP        WN 


ir.0  ^.  W  is  the  total  mean  pressure. 

L 


The  formula  then  becomes: 


138  MACHINE  DESIGN 

The  value  of  B  was  found  to  be  2.5  in.  for  high-speed  and  2  in. 
for  low-speed  engines,  while  K  fluctuated  from  .13  to  .46  with 
an  average  of  .30  in  the  former  class,  and  from  .40  to  .80  with  an 
average  of  .60  in  the  low-speed  engines. 

If  we  adopt  average  values  we  have  the  following  formulas 
for  the  crank  pins  of  modern  stationary  engines: 

WN 
High-speed  engines  I  =  66Q  Q()()  +  2.5  in.  (73) 

WN 
Low-speed  engines  I  =  +2  in- 


Compare  these  formulas  with  (71)  when  values  of  C  are 
introduced. 

In  a  discussion  on  the  subject  of  journal  bearings  in  1885,  1 
Mr.  Geo.  H.  Babcock  said  that  he  had  found  it  practicable  to 
allow  as  high  as  1200  Ib.  per  square  inch  on  crank  pins  while  the 
main  journal  could  not  carry  over  300  Ib.  per  square  inch  without 
heating.  One  rule  for  speed  and  pressure  of  journal  bearings 
used  by  a  well-known  designer  of  Corliss  engines  is  to  multiply 
the  square  root  of  the  speed  in  feet  per  second  by  the  pressure 
per  square  inch  of  projected  area  and  limit  this  product  to  350 
for  horizontal  engines  and  500  in  vertical  engines. 

62.  Experiments.  —  Some  tests  made  on  a  steel  journal  3J  in. 
in  diameter  and  8  in.  long  running  in  a  cast-iron  bearing  and 
lubricated  by  a  sight-feed  oiler,  will  serve  to  illustrate  the  friction 
and  heating  of  such  journals. 

The  two  halves  of  the  bearing  were  forced  together  by  helical 
springs  with  a  total  force  of  1400  Ib.,  so  that  there  was  a  pressure 
of  54  Ib.  per  square  inch  on  each  half.  The  surface  speed  was 
430  ft.  per  minute  and  the  oil  was  fed  at  the  rate  of  about  12 
drops  per  minute.  The  lubricant  used  was  a  rather  heavy 
automobile  oil  having  a  specific  gravity  of  0.925  and  a  viscosity 
of  174  when  compared  with  water  at  20°  Cent. 

The  length  of  the  run  was  two  hours  and  the  temperature  of 
the  room  70°  fahr.  (See  Table  XXXVII.) 

1  Trans.  A.  S.  M.  E.,  Vol.  VI. 


FRICTION  OF  JOURNALS 


139 


TABLE  XXXVII 
FRICTION  OF  JOURNAL  BEARING 


Time 

Rev.  per  min. 

Temp.  fahr. 

Coeff.  of  friction 

10:03 

500 

69 

.024 

10:15 

482 

82 

.0175 

10:30 

506 

100 

.013 

10:45 

506 

115 

.010 

11:00 

516 

125 

.010 

11  :15 

135 

004 

11  :30 

145 

004 

11  :45 

512 

147 

.004 

12:00 

.... 

151 

.007 

Mr.  Albert  Kingsbury  of  the  Westinghouse  Electric  and 
Manufacturing  Company  reports  some  valuable  experiments  on 
bearings  of  unusually  large  size  and  under  extremely  heavy 
pressures.1 

The  bearings  were  three  in  number;  two,  9  in.  in  diameter 
and  30  in.  long,  supporting  the  shaft,  and  one  15  in.  in  diameter 
and  40  in.  long  to  which  the  pressure  was  applied.  These 
bearings  are  designated  as  A,  B  and  C,  B  being  the  larger  one. 
The  bearings  were  lined  with  genuine  Babbitt  metal  and  scraped 
to  fit  the  shaft.  They  were  flooded  with  oil  from  a  supply  tank 
to  which  the  oil  was  returned  by  a  pump. 

The  runs  were  usually  of  about  seven  hours'  duration  and 
started  with  all  the  parts  cool. 

1  Trans.  A.  S.  M.  E,  Vol.  XXVII.v 


140 


MACHINE  DESIGN 


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142  MACHINE  DESIGN 

63.  Strength  and  Stiffness  of  Journals. — A  journal  is  usually 
in  the  condition  of  a  bracket  with  a  uniform  load,  and  the  bending 

*  ™     Wl 
moment  M-- — 

Therefore  by  formula  (6) 


=  3/102M      ./5.1TTZ 
"  \     tf~       \ !o~~ 


or  d=  1.721  V^-  (75) 


The  maximum  deflection  of  such  a  bracket  is 

Wl3 


A  = 


SEI 
Wl3 


64     8#A 
64W     2.547W13 


If  as  is  usual  A  is  allowed  to  be  T^  in.,  then  for  stiffness 

E—  w 

or  approximately  d  =  4  \/-™  (77) 

E 

The  designer  must  be  guided  by  circumstances  in  determining 
whether  the  journal  shall  be  calculated  for  wear,  for  strength 
or  for  stiffness.  A  safe  way  is  to  design  the  journal  by  the 
formulas  for  heating  and  wear  and  then  to  test  for  strength  and 
deflection. 

Remember  that  no  factor  of  safety  is  needed  in  formula  for 
stiffness. 

Note  that  W  in  formulas  for  strength  and  stiffness  is  not  the 
average  but  the  maximum  load. 

64.  Caps  and  Bolts. — The  cap  of  a  journal  bearing  exposed  to 
upward  pressure  is  in  the  condition  of  a  beam  supported  by  the 
holding  down  bolts  and  loaded  at  the  center,  and  may  be  designed 
either  for  strength  or  for  stiffness. 


JOURNALS  143 

Let:  P  =  max.  upward  pressure  on  cap  0 

L  =  distance  between  bolts 
b=  breadth  of  cap  at  center 
h  =  depth  of  cap  at  center 
A  —greatest  allowable  deflection. 

Sbh2     PL 
Strength:  Af=— __  =  __ 

/Q~PT 

(78) 

\    ALftO 

TT7  T  3 

Stiffness:  A  = 


(79) 

If  A   is  allowed  to  be  T|ir  in.  and  E  for  cast  iron  is  taken 
18,000,000 


then:  fc=.01115L      —  .  (80) 

The  holding  down  bolts  should  be  so  designed  that  the  bolts 
on  one  side  of  the  cap  may  be  capable  of  carrying  safely  two- 
thirds  of  the  total  pressure. 

PROBLEMS 

1.  A  flat  car  weighs  20  tons,  is  designed  to  carry  a  load  of  40  tons  more 
and  is  supported  by  two  four-wheeled  trucks,  the  axle  journals  being  of 
wrought  iron  and  the  wheels  33  in.  in  diameter. 

Design  the  journals,  considering  heating,  wear,  strength  and  stiffness, 
assuming  a  maximum  speed  of  30  miles  an  hour,  factor  of  safety  =  10  and 
C  =  300,000. 

2.  The  following  dimensions  are  those  generally  used  for  the  journals  of 
freight  cars  having  nominal  capacities  as  indicated: 

V 

Capacity  Dimensions  of  journal 

100,000  Ib  ...................................     4.  5  by  9  in. 

60,000  Ib  ...................................   4  .25  by  8  in. 

40,000  Ib  ...................................   3.75  by  7  in. 


144  MACHINE  DESIGN 

Assuming  the  weight  of  the  car  to  be  40  per  cent  of  its  carrying  capacity 
in  each  instance,  determine  the  pressure  per  square  inch  of  projected  area 
and  the  value  of  the  constant  C  { Formula  (71)}. 

3.  Measure  the  crank  pin  of  any  modern  engine  which    is  accessible, 
calculate  the  various  constants  and  compare  them  with  those  given  in  this 
chapter. 

4.  Design  a  crank  pin  for  an  engine  under  the  following  conditions: 

Diameter  of  piston  =28  in. 

Maximum  steam  pressure  =90  Ib.  per  square  inch. 

Mean  steam  pressure  =40  Ib.  per  square  inch. 

Revolutions  per  minute  =  75. 

Determine  dimensions  necessary  to  prevent  wear  and  heating  and  then 
test  for  strength  and  stiffness. 

5.  Design  a   crank  pin  for  a  high-speed  engine  having  the  following 
dimensions  and  conditions: 

Diameter  of  piston .  =  14  in. 

Maximum  steam  pressure  =  100  Ib.  per  square  inch. 

Mean  steam  pressure  =50  Ib.  per  square  inch. 

Revolutions  per  minute  =250. 

6.  Make  a  careful  study  and  sketch  of  journals  and  journal  bearings  on 
each  of  the  following  machines  and  analyze  as  to  (a)  Materials,     (b)  Adjust- 
ment,    (c)  Lubrication. 

a.  An  engine  lathe. 

b.  A  milling  machine. 

c.  A  steam  engine. 

d.  An  electric  generator  or  motor. 

7.  Sketch  at  least  two  forms  of  oil  cup  used  in  the  laboratories  and 
explain  their  working. 

8.  The  shaft  journal  of  a  vertical  engine  is  4  in.  in  diameter  by  6  in. 
long.     The  cap  is  of  cast  iron,  held  down  by  4  bolts  of  wrought  iron,  each, 
5  in.  from  center  of  shaft,  and  the  greatest  vertical  pressure  is  12,000  Ib. 

Calculate  depth  of  cap  at  center  for  both  strength  and  stiffness,  and  also 
the  diameter  of  bolts. 

9.  Investigate  the  strength  of  the  cap  and  bolts  of  some  pillow  block 
whose  dimensions  are  known,  under  a  pressure  of  500  Ib.  per  square  inch  of 
projected  area. 

10.  The  total  weight  on  the  drivers  of  a  locomotive  is  64,000  Ib.     The 
drivers  are  four  in  number,  5  ft.  2  in.  in  diameter,  and  have  journals  1\  in. 
in  diameter. 

Determine  horse-power  consumed  in  friction  when  the  locomotive  is 
running  50  miles  an  hour,  assuming  tan  ^  =  .05. 

65.  Step -Bearings. — Any  bearing  which  is  designed  to  resist 
end  thrust  of  the  shaft  rather  than  lateral  pressure  is  denomi- 


STEP  BEARINGS  145 

nated  a  step  or  thrust  bearing.  These  are  naturally  most  used 
on  vertical  sh'afts,  but  may  be  frequently  seen  on  horizontal 
ones  as  for  example  on  the  spindles  of  engine  lathes,  boring 
machines  and  milling  machines. 

Step-bearings  may  be  classified  according  to  the  shape  of  the 
rubbing  surface,  as  flat  pivots  and  collars,  conical  pivots,  and 
conoidal  pivots  of  which  the  Schiele  pivot  is  the  best  known. 
When  a  step-bearing  on  a  vertical  shaft  is  exposed  to  great  pres- 
sure or  speed  it  is  sometimes  lubricated  by  an  oil  tube  coming  up 
from  below  to  the  center  of  the  bearing  and  connecting  with  a 
stand  pipe  or  force-pump.  The  oil  entering  at  the  center  is 
distributed  by  centrifugal  force. 

66.  Friction  of  Pivots  or  Step -bearings. — Flat  Pivots. 

Let   W  =  weight  on  pivot 

d^  =  outer  diameter  of  pivot 
p  =  intensity  of  vertical  pressure 
T  =  moment  of  friction 
/=  coefficient  of  friction  =  tan  <p. 

We  will  assume  p  to  be  a  constant  which  is  no  doubt  approxi- 
mately true. 

W 

Then  v  = = 

area 

JL 

Let  r  =  the  radius  of  any  elementary  ring  of  a  width  =dr, 

then  area  of  element  =  2  nrdr 
Friction  of  element  =fpX 2 nrdr 
Moment  of  friction  of  element  =  2fpnr2dr 
and 

T  =  2fpn  f-  r2dr  (a) 


24 


The  great  objection  to  this  form  of  pivot  is  the  uneven  wear 
due  to  the  difference  in  velocity  between  center  and  circum- 
ference. 


146  MACHINE  DESIGN 

67.  Flat  Collar. 

Let  d  2  =  inside  diameter 
Integrating  as  in  equation  (a)  above,  but  using  limits 

y  and  -~  we  have 


In  this  case 

4TF 


and 

T-*w$=%  (82) 

68.  Conical  Pivot. 

Let  a  =  angle  of  inclination  to  the  vertical. 
As  in  the  case  of  a  flat  ring  the  intensity  of  the  vertical  pres- 
sure is 

4TF 
P    n(dl-dl) 

and  the  vertical  pressure  on  an  elementary  ring  of  the  bearing 
surface  is 


s 


- 

As  seen  in  Fig.  60  the  normal  pressure  on  the  elementary  ring 

dp-dW  =       SWrdr 
~ 


sina     (dl  —  d\]  sina 


STEP  BEARINGS  147 

The  friction  on  the  ring  isfdP  and  the  moment  of  this  friction  is 

-    AT.frip.Sy™ 

— 


r<L 

r-      *££_      a 

(dl-dl)sina   I   d, 
J     2 


As  a  approaches  ^  the  value  of  T  approaches  that  of  a  flat  ring, 

and  as  a  approaches  0  the  value  of  T  approaches  °o . 
If  d2  =  Q  we  have 

T  =  \ — -. —  (84) 

3  sina 

The  conical  pivot  also  wears  unevenly,  usually  assuming  a 
concave  shape  as  seen  in  profile. 

69.  Schiele's  Pivot. — By  experimenting  with  a  pivot  and 
bearing  made  of  some  friable  material,  it  was  shown  that  the 
outline  tended  to  become  curved  as  shown  c 
in  Fig.  62.  This  led  to  a  mathematical  in-  p 
vestigation  which  showed  that  the  curve 
would  be  a  tractrix  under  certain  conditions. 

This  curve  may  be  traced  mechanically  as 
shown  in  Fig.  61. 

Let  the  weight  W  be  free  to  move  on  a 
plane.     Let  the  string  SW  be  kept  taut  and   *         F 
the  end  S  moved  along  the  straight  line  SL. 
Then  will  a  pencil  attached  to  the  center  of  W  trace  on  the 
plane  a  tractrix  whose  axis  is  SL. 

In  Fig.  62.  let  SW  =  length  of  string  =  r  1  and  let  P  be  any  point 
in  the  curve.  Then  it  is  evident  that  the  tangent'PQ  to  the  curve 
is  a  constant  and  =  r1 

Also 

sinO 


148 


MACHINE  DESIGN 


Let  a  pivot  be  generated  by  revolving  the  curve  around  its 
axis  SL.  As  in  the  case  of  the  conical  pivot  it  can  be  proved 
that  the  normal  pressure  on  an  element  of  convex  surface  is 

SWrdr 


~ 


-  d     sinO 


S 


<-— r—  •*, 


FIG.  62. 


Let  the  normal  wear  of  the  pivot  be 
assumed  to  be  proportional  to  this  nor- 
mal pressure  and  to  the  velocity  of  the 
rubbing  surfaces,  i.e.  normal  wear  pro- 
portional to  pr,  then  is  the  vertical 
pr  „  r 


wear  proportional  to 


But 


sinO  J   sinO 

a  constant,  therefore  the  vertical  wear 
will  be  the  same  at  all  points.     This 
is  the  characteristic  feature  and  advan- 
tage of  this  form  of  pivot. 
As  shown  in  equation  (a) 


T  is  thus  shown  to  be  independent  of  d2  or  of  the  length  of 
pivot  used. 

This  pivot  is  sometimes  wrongly  called  antifriction.  As  will 
be  seen  by  comparing  equations  (81)  and  (85)  the  moment  of 
friction  is  50  per  cent  greater  than  that  of  the  common  flat 
pivot. 

The  distinct  advantage  of  the  Schiele  pivot  is  in  the  fact 
that  it  maintains  its  shape  as  it  wears  and  is  self-adjusting.  It 
is  an  expensive  bearing  to  manufacture  and  is  seldom  used  on 
that  account. 

It  is  not  suitable  for  a  bearing  where  most  of  the  pressure  is 
sideways. 

Mr.  H.  G.  Reist  of  the  General  Electric  Company,  in  the  paper 
before  alluded  to,  explains  the  practice  of  that  company  in 
regard  to  large  step  bearings  for  steam  turbine  work. 


STEP  BEARINGS 


149 


The  pressures  and  speeds  allowed  are  the  same  $s  already 
quoted  for  cylindrical  bearings.  The  bearings  are  usually  sub- 
merged in  oil  and  are  provided  with  radial  grooves  in  the  step 
journal  to  force  the  oil  over  the  entire  surface. 

Two  bearings  are  sometimes  employed,  one  above  the  other, 
one  being  supported  by  a  spring  so  as  to  take  about  one-half 
the  load. 

If  pressure  and  speed  are  great,  the  weight  is  supported  on  a 
film  of  oil  or  water  maintained  by  pressure.  A  circular  recess 
about  half  the  diameter  of  the  bearing  disc  allows  the  oil  to 
distribute.  The  distance  that  the  bearing  is  raised  by  the  oil 
pressure  is  from  .003  to  .005  in.  and  the  pressures  employed 
vary  from  250  to  800  Ib.  per  square  inch  according  to  circum- 
stances. The  initial  pressure  to  raise  the  step  will  be  about  25 
per  cent  greater.  The  following  figures  are  quoted  as  examples 
of  ordinary  practice. 


Weight  of  rotor  
Revolutions  per  minute 

9,800 
1  800 

53,000 
750 

187,000 
500 

Diameter  of  bearing  
Pressure  of  oil  .  . 

9.75 
180 

16 
420 

22.5 
650 

Quantity  of  oil  in  gallons  per  minute.  . 

1 

3.5 

70.  Multiple  Bearings. — To  guard  against  abrasion  in  flat 
pivots  a  series  of  rubbing  surfaces  which  divide  the  wear  is  some- 
times provided.  Several  flat  discs  placed  beneath  the  pivot  and 
turning  indifferently  may  be  used.  Sometimes  the  discs  are 
made  alternately  of  a  hard  and  a  soft  material.  Bronze,  steel 
and  raw  hide  are  the  more  common  materials. 

Notice  in  this  connection  the  button  or  washer  at  the  outer  end 
of  the  head  spindle  of  an  engine  lathe  and  the  loose  collar  on  the 
main  journal  of  a  milling  machine.  See  Figs.  54  and  55.  Pivots 
are  usually  lubricated  through  a  hole  at  the  center  of  the  bearing 
and  it  is  desirable  to  have  a  pressure  head  on  the  oil  to  force  it  in. 

The  hydraulic  foot  step  sometimes  used  for  the  vertical  shafts 
of  turbines  is  in  effect  a  rotating  plunger  supported  by  water 
pressure  underneath  and  so  packed  in  its  bearing  as  to  allow  a 


150 


MACHINE  DESIGN 


slight  leakage  of  water  for  cooling  and  lubricating  the  bearing 
surfaces. 

The  compound  thrust  bearing  generally  used  for  propeller 
shafts  consists  of  a  number  of  collars  of  the  same  size  forged  on 
the  shafts  at  regular  intervals  and  dividing  the  end  thrust  between 
them,  thus  reducing  the  intensity  of  pressure  to  a  safe  limit 
without  making  the  collars  unreasonably  large. 

Fig.  63  shows  the  shape  of  the  horseshoe  rings  for  bearing 
surface  arranged  for  independent  water  cooling. 


FIG.  63. 


A  safe  value  for  p  the  intensity  of  pressure  is,  according  to 
Whitham,  60  Ib.  per  square  inch  for  high-speed  engines. 

A  table  given  by  Prof.  Jones  in  his  book  on  Machine  Design 
shows  the  practice  at  the  Newport  News  ship-yards  on  marine 
engines  of  from  250  to  5000  horse-power.  The  outer  diameter 
of  collars  is  about  one  and  one-half  times  the  diameter  of  the 
shafts  in  each  case  and  the  number  of  collars  used  varies  from 
6  in  the  smallest  engine  to  11  in  the  largest.  The  pressure  per 
square  inch  of  bearing  surface  varies  from  18  to  46  Ib.  with  an 
average  value  of  about  32  Ib. 

Mr.  G.  W.  Dickie  gives  some  data  concerning  modern  naval 
practice  in  the  design  of  thrust  bearings.  The  usual  method  of 
determining  the  pressure  is  to  assume  two-thirds  of  the  indi- 
cated horse-power  and  calculate  the  pressure  from  that  by 
the  formula: 


THRUST  BEARINGS 


151 


=  l  HP 


2X60X33000 
3X6080S 


(86) 


where  P  =  pressure  on  thrust  bearing  S  =  speed  of  ship  in 
knots. 

Mr.  Dickie  quotes  examples  from  modern  practice  for  both 
naval  and  merchant  service.  These  are  assembled  in  Table 
XXXIX. 

All  of  these  bearings  except  No.  4  were  supplied  with  water  cir- 
culation through  each  horseshoe.  No.  3  required  especial  care 
when  running  on  account  of  the  high  rubbing  velocity. 

TABLE  XXXIX 

PROPERTIES  OF  MARINE  THRUST  BEARINGS  (DICKIE) 


1 

2 

3 

4 

Data 

Armored 

Protected 

Torpedo  boat 

Passenger 

cruiser 

cruiser 

destroyer 

steamer 

Speed,  knots  

22 

22.5 

28 

21 

Surface  of  ring  (square  inch)  

1,188 

891 

581 

2,268 

Horse-power,  one  engine  

11,500 

6,800 

4,200 

15,000 

Total  thrust  (pounds)  

112,700 

89,000 

33,600 

154,500 

Pressure  per  square  inch  (pounds)  . 

95 

100 

58 

68.1 

Mean    rubbing    speed    (feet    per 

642 

610 

827 

504 

minute). 

PROBLEMS 

1.  Design  and  draw  to  full  size  a  Schiele  pivot  for  a  water  wheel  shaft 
4  in.  in  diameter,  the  total  length  of  the  bearing  being  3  in. 

Calculate  the  horse-power  expended  in  friction  if  the  total  vertical 
pressure  on  the  pivot  is  two  tons  and  the  wheel  makes  150  revolutions  per 
minute  and  assuming /=. 25  for  metal  on  wet  wood. 

2.  Compare  the  friction  of  the  pivot  in  Prob.  1,  with  that  of  a  flat  collar 
of  the  same  projected  area  and  also  with  that  of  a  conical  pivot  having 
a  =30  degrees. 

3.  Design  a  compound  thrust  bearing  for  a  propeller  shaft  the  diameters 
being  14  and  21  in.,  the  total  thrust  being  80,000  Ib.and  the  pressure^  60  Ib. 
per  square  inch. 

Calculate  the  horse-power  consumed  in  friction  and  compare  with  that 
developed  if  a  single  collar  of  same  area  had  been  used.  Assume  /=  .05 
and  revolutions  per  minute  =  120. 


152  MACHINE  DESIGN 


REFERENCES 

Machine  Design.     Low  and  Bevi?,  Chapter  IX. 

Steam  Engine.     Ripper,  Chapter  XVII. 

Large  Experimental  Bearing.     Am.  Mach.,  March  15,  1906. 

Experiments  on  Bearings.     Am.  Mach.,  Oct.  18,  1906. 

Lubrication  of  Bearings.     Tr.  A.  S.  M.  E.,  Vol.  X,  p.  810;  Vol.  XIII, 

p.  374. 

Lubrication.     Eng.  Mag.,  Dec.,  1908. 
Bearings,  a  Symposium.     Tr.  A.  S.  M.  E.,  Vol.  XXVII,  p.  420. 


CHAPTER  VIII 


BALL  AND  ROLLER  BEARINGS 

71.  General  Principles. — The  object  of  interposing  a  ball  or 
roller  between  a  journal  and  its  bearing,  is  to  substitute  rolling 
for  sliding  friction  and  thus  to  reduce  the  resistance.  This  can 
be  done  only  partially  and  by  the  observance  of  certain  principles. 
In  the  first  place  it  must  be  remembered  that  each  ball  can  roll 
about  but  one  axis  at  a  time;  that  axis  must  be  determined  and 
the  points  of  contact  located  accordingly. 

Secondly,  the  pressure  should  be  approximately  normal  to  the 
surfaces  at  the  points  of  contact. 

Finally  it  must  be  understood,  that  on  account  of  the  contact 
surfaces  being  so  minute,  a  comparatively  slight  pressure  will 
cause  distortion  of  the  balls  and  an  entire  change  in  the 
conditions. 


may  be  either  two,   three  or 


FIG.  64. 


72.  Journal  Bearings. — These 
four  point,  so  named  from 
the  number  of  points  of  con- 
tact of  each  ball.  r 

The  axis  of  the  ball  may  A 
be  assumed  as  parallel  or  in-  ^ 
clined  to  the  axis  of  the  jour- 
nal and  the  points  of  contact 
arranged  accordingly.  The 
simplest  form  consists  of  a  plain  cylindrical  journal  running 
in  a  bearing  of  the  same  shape  and  having  rings  of  balls  inter- 
posed. The  successive  rings  of  balls  should  be  separated 
by  thin  loose  collars  to  keep  them  in  place.  These  collars  are 
a  source  of  rubbing  friction,  and  to  do  away  with  them  the 
balls  are  sometimes  run  in  grooves  either  in  journal,  bearing  or 
both. 

Fig.  64  shows  a  bearing  of  this  type,  there  being  three  points 
of  contact  and  the  axis  of  ball  being  parallel  to  that  of  journal. 

153 


154 


MACHINE  DESIGN 


•D 


The  bearings  so  far  mentioned  have  no  means  of  adjustment 
for  wear.  Conical  bearings,  or  those  in  which  the  axes  of  the 
balls  meet  in  a  common  point,  supply  this  deficiency.  In 
designing  this  class  of  bearings,  either  for  side  or  end  thrust,  the 
inclination  of  the  axis  is  assumed  according  to  the  obliquity 
desired  and  the  points  of  contact  are  then  so  located  that  there 
shall  be  no  slipping. 

Fig.  65  illustrates  a  common  form  of  adjustable  or  cone  bearing 
and  shows  the  method  of  designing  a  three-point  contact.  A  C 
is  the  axis  of  the  cone,  while  the  shaded  area  is  a  section  of  the 
cup,  so  called.  Let  a  and  b  be  two  points  of  contact  between 

ball  and  cup.  Draw  the  line  a  b 
and  produce  to  cut  axis  in  A. 
Through  the  center  of  ball  draw 
the  line  A  B;  then  will  this  be 

^  the    axis  of  rotation  of  the  ball 

JL — A&.-™.  .-JL Q_£.    and  a  c,  b  d  will  be  the  projections 

U of  two  circles  of  rotation.     As  the 

radii  of  these  circles  have  the 
same  ratio  as  the  radii  of  revolu- 
tion a  n,  b  m,  there  will  be  no 
slipping  and  the  ball  will  roll  as  a 
cone  inside  another  cone.  The 

exact  location  of  the  third  point  of  contact  is  not  material.  If 
it  were  at  c,  too  much  pressure  would  come  on  the  cup  at  6;  if 
at  d  there  would  be  an  excess  of  pressure  at  a,  but  the  rolling 
would  be  correct  in  either  case.  A  convenient  method  is  to 
locate  p  by  drawing  A  D  tangent  to  ball  circle  as  shown.  It  is 
recommended,  however,  that  the  two  opposing  surfaces  at  p 
and  b  or  a  should  make  with  each  other  an  angle  of  not  less  than 
25  degrees  to  avoid  sticking  of  the  ball. 

To  convert  the  bearing  just  shown  to  four-point  contact,  it 
would  only  be  necessary  to  change  the  one  cone  into  two  cones 
tangent  to  the  ball  at  c  and  d. 

To  reduce  it  to  two-point  contact  the  points  a  and  b  are 
brought  together  to  a  point  opposite  p.  As  in  this  last  case  the 
ball  would  not  be  confined  to  a  definite  path  it  is  customary  to 
make  one  or  both  surfaces  concave  conoids  with  a  radius  about 
three-fourths  the  diameter  of  the  ball.  See  Fig.  66. 


FIG.  65. 


BALL  BEARINGS 


155 


73.  Step -bearings. — The  same  principles  apply  as*  in  the  pre- 
ceding article  and  the  axis  and  points  of  contact  may  be  varied 
in  the  same  way.  The  most  common  form  of  step-bearing  con- 
sists of  two  flat  circular  plates  separated  by  one  or  more  rings  of 
balls.  Each  ring  must  be  kept  in  place  by  one  or  more  loose 
retaining  collars,  and  these  in  turn  are  the  cause  of  some  sliding 


friction.  This  is  a  bearing  with  two-point  contact  and  the  balls 
turning  on  horizontal  axes.  If  the  space  between  the  plates  is 
filled  with  loose  balls,  as  is  sometimes  done,  the  rubbing  of  the 
balls  against  each  other  will  cause  considerable  friction. 

To  guide  the  balls  without  rubbing  friction  three-point  contact 
is  generally  used. 


FIG.  68. 

Fig.  67  illustrates  a  bearing  of  this  character.  The  method 
of  design  is  shown  in  the  figure,  the  principle  being  the  same  as 
in  Fig.  65.  By  comparing  the  lettering  of  the  two  figures  the 
similarity  will  be  readily  seen. 


156  MACHINE  DESIGN 

This  last  bearing  may  be  converted  to  four-point  contact  by 
making  the  upper  collar  of  the  same  shape  as  the  lower. 

What  is  practically  a  two-point  contact  with  some  of  the 
advantages  of  four  point  is  attained  by  the  use,  of  curved  races  for 
the  balls  as  in  Fig.  68. 

To  insure  even  distribution  of  the  load,  the   lower  ring  is 

supported  on  a  self-adjusting  spher- 
ical collar.  The  radii  of  the  curved 
races  should  not  be  less  than  two- 
thirds  the  diameter  of  the  balls. 

To  guide  the  balls  in  two-point 
contact  use  is  sometimes  made  of  a 
cage  ring,  a  flat  collar  drilled  with 
holes  just  a  trifle  larger  than  the 
balls  and  disposing  them  either  in 
spirals  or  in  irregular  order.  See 
Fig.  69. 

This  method  has  the  advantage  of 

making  each  ball  move  in  a  path  of  different  radius  thus  secur- 
ing more  even  wear  for  the  plates. 

74.  Materials  and.  Wear. — The  balls  themselves  are  always 
made  of  steel,  hardened  in  oil,  tempered  and  ground.  They 
are  usually  accurate  to  within  one  ten-thousandth  of  an  inch. 
The  plates,  rings  and  journals  must  be  hardened  and  ground 
in  the  same  way  and  perhaps  are  more  likely  to  wear  out 
or  fail  than  the  balls.  A  long  series  of  experiments  made  at 
the  Case  School  of  Applied  Science  on  the  friction  and 
endurance  of  ball  step-bearings  showed  some  interesting 
peculiarities. 

Using  flat  plates  with  one  circle  of  quarter-inch  balls  it  was 
found  that  the  balls  pressed  outward  on  the  retaining  ring  with 
such  force  as  to  cut  and  indent  it  seriously.  This  was  probably 
due  to  the  fact  that  the  pressure  slightly  distorted  the  balls  and 
changed  each  sphere  into  a  partial  cylinder  at  the  touching  points. 
While  of  this  shape  it  would  tend  to  roll  in  a  straight  line  or  a 
tangent  to  the  circle.  Grinding  the  plates  slightly  convex  at  an 
angle  of  1  to  1£  degrees  obviated  the  difficulty  to  a  certain 
extent.  Under  even  moderately  heavy  loads  the  continued 


BALL  BEARINGS  157 

rolling  of  the  ring  of  balls  in  one  path  soon  damaged  the  plates 
to  such  an  extent  as  to  ruin  the  bearing. 

A  flat  bearing  filled  with  loose  balls  developed  three  or  four 
times  the  friction  of  the  single  ring  and  a  three-point  bearing 
similar  to  that  in  Fig.  68  showed  more  than  twice  the  friction  of 
the  two  point  bearing. 

A  flat  ring  cage  such  as  has  already  been  described  was  the 
most  satisfactory  as  regards  friction  and  endurance. 

The  general  conclusions  derived  from  the  experiments  were 
that  under  comparatively  light  pressures  the  balls  are  distorted 
sufficiently  to  disturb  seriously  the  manner  of  rolling  and  that 
it  is  the  elasticity  and  not  the  compressive  strength  of  the  balls 
which  must  be  considered  in  designing  bearings. 

75.  Design    of    Bearings. — Figures    on    the    direct    crushing 
strength  of  steel  balls  have  little  value  for  the  designer.     For 
instance  it  has  been  proved  by  numerous  tests  that  the  average 
crushing  strengths  of  ^-in.  and  f-in.  balls  are  about  7500  Ib. 
and  15,000  Ib.  respectively.     Experiments  made  by  the  writer 
show  that  a  J-in.  ball  loses  all  value  as  a  transmission  element 
on  account  of  distortion,  at  any  load  of  more  than  100  Ib. 

Prof.  Gray  states,  as  a  conclusion  from  some  experiments 
made  by  him,  that  not  more  than  40  Ib.  per  ball  should  be  allowed 
for  f-in.  balls. 

This  distortion  doubtless  accounts  for  the  failure  of  theoretic- 
ally correct  bearings  to  behave  as  was  expected  of  them. 

Mr.  Charles  R.  Pratt  reports  the  limit  of  work  for  J-in.  balls 
in  thruet  bearings  to  be  100  Ib.  per  ball  at  700  revolutions  per 
minute  and  6  in.  diameter  circle  of  rotation. 

Mr.  W.  S.  Rogers  gives  the  maximum  load  for  a  1-in.  ball  as 
1000  Ib.  and  for  a  J-in.  ball  as  200  Ib. 

76.  Endurance  of  Ball  Bearings. — For  complete  and  reliable 
data  on  the  strength  and  endurance  of  ball  bearings,  reference 
is  made  to  a  paper  by  Mr.  Henry  Hess  and  to  translations  of  the 
work  of  Professor  Stribeck.1 

The  formulas  which  follow  are  derived  mainly  from  the  sources 
mentioned. 

1  Trans.  A.  S.  M.  E.,  Vol.  XXIX. 


158  MACHINE  DESIGN 

Ball  bearings  do  not  fail  from  wear  but,  as  already  noticed, 
from  distortion  and  injury  at  the  contact  points.     The  use  of  a 
curved  race,  as  in  Fig.  68,  will  increase  the  durability,  because 
the  contact  point  is  reinforced  by  the  material  at  either  side. 
In  a  journal  bearing  having  one  ring  of  balls,  one-fifth  of  the 
total  number  of  balls  is  considered  as  carrying  the  load.     In  a 
plain  journal,  the  unit  is  the  square  inch  of  projected  area.     In 
a  ball  bearing,  for  projected  area  is  substituted  the  square  of 
ball  diameter  multiplied  by  one-fifth  the  number  of  balls. 
Let     d  =  diameter  of  ball  in  inches 
n  =  number  of  balls  in  ring 
W  =  total  load  on  balls 
=  safe  load  on  one  ball. 


Then     p  =  --  =  kd2  (a) 

n 

where  A;  is  a  constant  depending  on  the  material  and  the  type  of 
bearing. 

From  equation  (a)  : 

W-k?£  (87) 

o 

nd2 

where  —=-  corresponds  to  the  (Id)  or  projected  area  of  the  plain 
o 

bearing.     The  values  of  k  are  as  follows  : 


Shape  of  race 

Hardened  steel  balls 

Hardened  steel  alloy  balls 

Flat  

500  to  700 

700  to  1  000 

Curved  to  radius  =  f  d  . 

1,500 

2,000 

The  load  capacity  of  balls  may  be  affected  by  various  condi- 
tions; lack  of  uniformity  in  the  hardness  of  either  balls  or  race 
will  reduce  the  capacity;  lack  of  uniformity  in  size  of  balls  is 
also  a  source  of  inefficiency;  sudden  variations  of  speed  cause 
shocks  which  impair  the  capacity  of  the  bearing. 

The  ball  bearing  is  of  somewhat  the  same  nature  as  a  chain  or  a 
gear  and  weakness  of  any  unit  leads  to  the  destruction  of  the 
whole. 


ROLLER  BEARINGS  159 

The  average  coefficient  of  friction  of  a  good  ball  bearing  is 
about  0.0015. 

Speeds  of  over  1500  revolutions  per  minute  are  impracticable. 

77.  Roller  Bearings. — The  principal  disadvantage  of  ball 
bearings  lies  in  the  fact  that  contact  is  only  at  a  point  and  that 
even  moderate  pressure  causes  excessive  distortion  and  wear. 
The  substitution  of  cylinders  or  cones  for  the  balls  is  intended  to 
overcome  this  difficulty. 

The  simplest  form  of  roller  bearing  consists  of  a  plain  cylindrical 
journal  and  bearing  with  small  cylindrical  rollers  interposed 
instead  of  balls.  There  are  two 
difficulties  here  to  be  overcome. 


The   rollers  tend  to  work  end-     ||  A 

ways  and  rub  or  score  whatever      y          / 

retains  them.     They  also  tend 

to    twist    around    and    become  -,-,      7n 

.bio.  70. 

unevenly  worn  or  even  bent  and 

broken,  unless  held  in  place  by  some  sort  of  cage.  In  short  they 
will  not  work  properly  unless  guided  and  any  form  of  guide  en- 
tails sliding  friction.  The  cage  generally  used  is  a  cylindrical 
sleeve  having  longitudinal  slots  which  hold  the  rollers  loosely 
and  prevent  their  getting  out  of  place  either  sideways  or  endways. 
The  use  of  balls  or  convex  washers  at  the  ends  of  the  rollers 
has  been  tried  with  some  degree  of  success.  See  Fig.  70.  Large 

rollers  have  been  turned  smaller  at  the 
ends  and  the  bearings  then  formed 
allowed  to  turn  in  holes  bored  in  re- 
volving collars.  These  collars  must 
be  so  fastened  or  geared  together 
as  to  turn  in  unison. 

FlG  71<  78.    Grant    Roller    Bearing.— The 

Grant  roller  is  conical  and  forms  an 

intermediate  between  the  ball  and  the  cylindrical  roller  having 
some  of  the  advantages  of  each.  The  principle  is  much  the 
same  as  in  the  adjustable  ball  bearing,  Fig.  65,  rolling  cones 
being  substituted  for  balls,  Fig.  71.  The  inner  cone  turns 
loose  on  the  spindle.  The  conical  rollers  are  held  in  position 


160  MACHINE  DESIGN 

by  rings  at  each  end,  while  the  outer  or  hollow  cone  ring  is  ad- 
justable along  the  axis. 

Two  sets  of  cones  are  used  on  a  bearing,  one  at  each  end  to 
neutralize  the  end  thrust,  the  same  as  with  ball  bearings. 

79.  Hyatt  Rollers. — The  tendency  of  the  rollers  to  get  out  of 
alignment  has  been  already  noticed.  The  Hyatt  roller  is  in- 
tended by  its  flexibility  to  secure  uniform  pressure  and  wear 
under  such  conditions.  It  consists  of  a  flat  strip  of  steel  wound 
spirally  about  a  mandrel  so  as  to  form  a  continuous  hollow 
cylinder.  It  is  true  in  form  and  comparatively  rigid  against 
compression,  but  possesses  sufficient  flexibility  to  adapt  itself  to 
slight  changes  of  bearing  surface. 

Experiments  made  by  the  Franklin  Institute  show  that  the 
Hyatt  roller  possesses  a  great  advantage  in  efficiency  over  the 
solid  roller. 

Testing  f-in.  rollers  between  flat  plates  under  loads  increasing 
to  550  Ib.  per  linear  inch  of  roller  developed  coefficients  of 
friction  for  the  Hyatt  roller  from  23  to  51  per  cent  less  than  for 
the  solid  roller.  Subsequent  examination  of  the  plates  showed 
also  a  much  more  even  distribution  of  pressure  for  the  former. 

A  series  of  tests  were  conducted  by  the  writer  in  1904-05  to 
determine  the  relative  efficiency  of  roller  bearings,  as  compared 
with  plain  cast-iron  and  Babbitted  bearings  under  similar  con- 
ditions.1 The  bearings  tested  had  diameters  of  Ijf,  2^,  2-^, 
and  2ff  in.  and  lengths  approximately  four  times  the  diameters. 
In  the  first  set  of  experiments  Hyatt  roller  bearings  wrere  com- 
pared with  plain  cast-iron  sleeves,  at  a  uniform  speed  of  480 
revolutions  per  minute  and  under  loads  varying  from  64  to  264 
Ib.  The  cast-iron  bearings  were  copiously  oiled. 

As  the  load  was  gradually  increased,  the  value  of  /  the  coeffi- 
cient of  friction  remained  nearly  constant  with  the  plain  bearings, 
but  gradually  decreased  in  the  case  of  the  roller  bearings.  Table 
XL  gives  a  summary  of  this  series  of  tests. 

The  relatively  high  values  of  /  in  the  2T8^-  and  2||  roller  bearings 
were  due  to  the  snugness  of  the  fit  between  the  journal  and  the 
bearing,  and  show  the  advisability  of  an  easy  fit  as  in  ordinary 
bearings. 

1  Mchy.,  N.  Y.,  Oct.,  1905. 


ROLLER  BEARINGS 


161 


TABLE  XL 

COEFFICIENTS  OF  FKICTION  FOR  ROLLER  AND  PLAIN   BEARINGS 


Hyatt  bearing 

Plain  bearing 

Diameter  of 

journal 

1 

Max. 

Min, 

Ave. 

Max. 

Min. 

Ave. 

IB 

.036 

.019 

.026 

.160 

.099 

.117 

2& 

.052 

.034 

.040 

.129 

.071 

.094 

2iV 

.041 

.025 

.030 

.143 

.076 

.104 

2H 

.053 

.049 

.051 

.138 

.091 

.104 

The  same  Hyatt  bearings  were  used  in  the  second  set  of 
experiments,  but  were  compared  with  the  Me  Keel  solid  roller 
bearings  and  with  plain  Babbitted  bearings  freely  oiled.  The 
McKeel  bearings  contained  rolls  turned  from  solid  steel  and 
guided  by  spherical  ends  fitting  recesses  in  cage  rings  at  each 
end.  The  cage  rings  were  joined  to  each  other  by  steel  rods 
parallel  to  the  rolls.  The  journals  were  run  at  a  speed  of  560 
revolutions  per  minute  and  under  loads  varying  from  113  to 
456  Ib.  Table  XLI  gives  a  summary  of  the  second  series  of 
tests. 

TABLE  XLI 
COEFFICIENTS  OF  FRICTION  FOR  ROLLER  AND  PLAIN   BEARINGS 


Diam. 

Hyatt  bearing 

McKeel  bearing 

Babbitt  bearing 

r»f 

journal 

Max. 

Min. 

Ave. 

Max. 

Min. 

Ave. 

Max. 

Min. 

Ave. 

i* 

.032 

.012 

.018 

,033 

.017 

.022 

.074 

.029 

.043 

2^ 

019 

Oil 

014 

088 

.078 

.082 

2T7e 

.042 

.025 

.032 

.028 

.015 

.021 

.114 

.083 

.096 

211 

.029 

.022 

.025 

.039 

.019 

.027 

.125 

.089 

.107 

The  variation  in  the  values  for  the  Babbitted  bearing  is  due 
to  the  changes  in  the  quantity  and  temperature  of  the  oil.     For 


162 


MACHINE  DESIGN 


heavy  pressures  it  is  probable  that  the  plain  bearing  might  be 
more  serviceable  than  the  others.  Notice  the  low  values  for  / 
in  Table  XXXVII. 

Under  a  load  of  470  Ib.  the  Haytt  bearing  developed  an  end 
thrust  of  13.5  Ib.  and  the  McKeel  one  of  11  Ib. 

This  is  due  to  a  slight  skewing  of  the  rolls  and  varies,  some- 
times reversing  in  direction. 

If  roller  bearings  are  properly  adjusted  and  not  overloaded  a 
saving  of  from  two  thirds  to  three-fourths  of  the  friction  may  be 
reasonably  expected. 

Professor  A.  L.  Williston  reports  some  tests  of  Hyatt  roller 
bearings  made  at  Pratt  Institute  in  1904.  The  journals  were 
1.5  in.  diameter  and  4  in.  long.  The  speeds  varied  from  128  to 
585  revolutions  per  minute.  Both  the  roller  and  the  plain 
bearings  were  lubricated  with  the  same  grade  of  oil.  The  total 
load  on  the  bearing  was  gradually  increased  from  1900  Ib.  to 
8300  Ib.  The  average  friction  of  each  bearing  was  as  given  in  the 
table. 

TABLE  XLII 
COEFFICIENTS  OF  FRICTION  FOR  ROLLER  AND  PLAIN  BEARINGS 


Revolutions  per 
minute 

Hyatt           Plain  cast  iron 

Plain  bronze 

130 

.0114 

.0548 

.0576 

302-320 

.0099 

.0592 

.0661 

410-585 

.0147 

.0683 

.140 

In  several  instances  the  cast-iron  bearing  seized  under  pres- 
sures above  5000  Ib.  while  the  bronze  bearing  proved  unreliable  at 
pressures  over  3000  Ib. 

The  roller  bearing  was  further  tested  with  total  pressures 
from  10,800  Ib.  to  23,500  Ib.  at  215  revolutions  per  minute  and 
the  coefficient  found  to  vary  from  .0094  to  ,0076. 

80.  Roller  Step -bearings. — In  article  74  attention  was  called 
to  the  fact  that  the  balls  in  a  step-bearing  under  moderately 
heavy  pressures  tend  to  benorne  cylinders  or  cones  and  to  roll 


ROLLER  BEARINGS 


163 


accordingly.  This  has  suggested  the  use  of  small  corses  in  place 
of  the  balls,  rolling  between  plates  one  or  both  of  which  are  also 
conical.  A  successful  bearing  of  this  kind  with  short  cylinders 
in  place  of  cones  is  used  by  the  Sprague-Pratt  Elevator  Co.,  and 
is  described  in  the  American  Machinist  for  June  27,  1901.  The 
rollers  are  arranged  in  two  spiral  rows  so  as  to  distribute  the 
wear  evenly  over  the  plates  and  are  held  loosely  in  a  flat  ring 
cage.  This  bearing  has  run  well 
in  practice  under  loads  double 
those  allowable  for  ball  bearings, 
or  over  100  Ib.  per  roll  for  rolls 
i  in.  in  diameter  and  J  in.  long. 
Fig.  72  illustrates  a  bearing 
of  this  character.  Collars  simi- 
tar to  this  have  been  used  in 
thrust  bearings  for  propeller 
shafts. 


81.  Design  of  Roller  Bearings. 

— Further  reference  is  here  made 

to  the  discussion  mentioned  in  Art.  76  for  information  as  to  the 

design  and  construction  of  roller  bearings.     As  in  the  case  of 

ball  bearings,  one-fifth  of  all  the  rolls  is  assumed  to  carry  the 

load  and  the  area  used  for  comparison  may  be  expressed  by  the 

formula: 

nld 
5 
where, 

n  =  number  of  rolls 

d  =  diameter  of  rolls  in  inches 

Z=  length  of  rolls  in  inches. 
The  allowable  pressure  per  roll  is, 

p  =  kld  (a) 

where  k  is  a  constant  depending  on  the  material  and  shape  of  the 
roll.     The  whole  load  on  the  bearing  is 

W  =  knl*-  (88) 


Mr.  Frank  Mossberg  gives  the  following  values  of  the  safe  load 


164 


MACHINE  DESIGN 


for  roller  bearings  of  the  Mossberg  type.  These  bearings  have 
small  solid  steel  rolls  hardened  to  a  spring  temper  and  guided  by 
bronze  cages  similar  to  those  mentioned  in  Art.  77.  The  journal 
is  tempered  to  a  medium  hardness  and  the  box  is  of  high  carbon 
steel  and  very  hard. 

TABLE  XLIII 
SAFE  LOAD  ON  MOSSBERG  ROLLER  BEARINGS 


Length  of 
journal, 
inches 

Diameter 
of 
journal, 
inches 

Diameter 
of 
rolls, 
inches 

Number 
of 
rolls 

Safe  load 
on 
journal, 
pounds 

Value  of  k 
5W 
~  nld 

I 

d 

n 

W 

3 

2 

t 

20 

3,500 

1,170 

3.75 

2.5 

A 

22 

7,000 

1,350 

4.5 

3 

t 

22 

13,000 

1,750 

6 

4 

TV 

24 

24,000 

1,900 

7.5 

5 

T9* 

24 

37,000 

,830 

9 

6 

U 

24 

50,000 

,690 

10.5 

7 

it 

22- 

70,000 

,860 

12 

8 

I 

22 

90,000 

,950 

13.5 

9 

1 

24 

115,000 

,770 

18 

12 

H 

26 

175,000 

,500 

22.5 

15 

if 

28 

255,000 

,470 

27 

18 

if 

32 

325,000 

,370 

30 

20 

li 

34 

400,000 

,300 

36 

24 

14 

38 

576,000 

,400 

Average 

1,590 

It  will  be  noticed  that  k  is  not  constant  in  Table  XLIII,  being 
greatest  for  the  intermediate  sizes.     The  average  value  is  about 


A;  =  1600. 


Smith  and  Marx  give: 


k  =  1000  for  hardened  steel 
A;  =400  for  cast  iron. 


ROLLER  BEARINGS 


165 


Mr.  Mossberg  considers  one-third  the  entire  number  of  rolls 
as  bearing  the  load.     This  would  make  the  formula  read: 


nld 


(89) 


with  an  average  value  of  k  =  960. 

The  roller  step-bearings  of  the  same  manufacture  have  small 
conical  rolls  with  an  angle  of  not  over  6  or  7  degrees;  retaining  rings 
or  cages  keep  the  rolls  in  correct  positions. 

The  bearing  collars  are  of  very  hard  high  carbon  steel  and  the 
rolls  as  in  the  journal  bearing  have  a  medium  or  spring  temper. 
Table  XLIV  gives  the  proportions  and  safe  loads. 

TABLE  XLIV 

SAFE  LOADS  ON  MOSSBERG  ROLLER  STEP-BEARINGS 


Diameter 

Number 

Area  of 

Safe  load  in  pounds  =  W 

of  shaft, 

of 

collar, 

inches, 
D 

rolls, 
n 

square 
inches 

75  revolutions 
per  minute 

150  revolutions 
per  minute 

2.25 

30 

10 

19,000 

9,500 

3.25 

30 

20 

40,000                      20,000 

4.25 

30 

35 

70,000 

35,000 

5.25 

30 

54 

108,000 

56,000 

6.50 

30 

78 

125,000 

62,000 

8.50 

32 

132 

200,000 

100,000 

9.50 

32 

162 

300,000 

150,000 

The  formulas  for  pressure  on  rolls  would  be  the  same  as  in 
journal  bearings  except  that  the  full  number  of  rolls — n — would 
be  effective  at  all  times. 


=  knld 


(90) 


where  I  is  the  length  of  roll  and  d  is  its  mean  diameter. 

The  table  gives  no  information  as  to  the  proportions  of  the 
roll. 

The  following  proportions  are  scaled  from  a  cut  of  the  bearing: 


166 


MACHINE  DESIGN 


Angle  of  cone  about  7  degrees. 

Z  =  0.36.D         d  =  0.1D        ld  =  .Q36D2. 
where 

D  =  diameter  of  shaft 
£=  working  length  of  roll 
d  =  mean  diameter  of  roll. 

TABLE  XLV 

VALUES  OF  k  FOR  ROLLER  THRUST  BEARING 


W 

Values  o; 

k  =~ 

nld 

D 

I 

d 

Id 

nld 

75  revolutions 

150  revolutions 

per  minute 

per  minute 

2.25 

0.81 

.225 

0.18 

5.46 

3,490 

,745 

3.25 

1.17 

.325 

0.38 

11.4 

3,500 

,750 

4.25 

1.53 

.425 

0.65 

19.5 

3,590 

,795 

5.25 

1.89 

.525 

0.99 

29.7 

3,640 

,820 

6.50 

2.34 

.650 

1.52 

45.6 

2,740 

,370 

8.50 

3.06 

.850 

2.60 

83.3 

2,400 

,200 

9.50 

3.42 

.950 

3.25 

104.0 

2,880 

,440 

Space  forbids  reference  to  all  of  the  many  varieties  of  ball  and 
roller  bearings  shown  in  manufacturers'  catalogues.  These  are 
all  subject  to  the  laws  and  limitations  mentioned  in  this  chapter, 

While  such  bearings  will  be  used  more  and  more  in  the  future, 
it  must  be  understood  that  extremely  high  speeds  or  heavy 
pressures  are  unfavorable  and  in  most  cases  prohibitive. 

Furthermore,  unless  a  bearing  of  this  character  is  carefully 
designed  and  well  constructed  it  will  prove  to  be  worse  than 
useless. 

REFERENCES 

Tests  of  Roller  Bearings.     Mchy.,  Oct.,  1905. 

Tests  of  Ball  Bearings.     Am.  Mach.,  Mar.  15,  1906,  Jan.  23,  1908. 

Pressure  on  Balls.     Am.  Mach.,  July  12,  1906. 

Discussion  on  Ball  Bearings.     Power,  July,  1907. 

Design  of  Roller  Bearings.     Power,  Jan.  14,  1908. 

Recent  Progress  in  Ball  Bearings.     Am.  Mach.,  Nov.  7,  1907. 

Symposium.     Tr.  A.  S.  M.  E.,  Vol.  XXVII,  p.  442. 

A  Very  Complete  Discussion.     Tr.  A.  S.  M.  E.,  Vol.  XXIX,  p.  367. 


CHAPTER  IX 
SHAFTING,  COUPLINGS  AND  HANGERS 

82.  Strength  of  Shafting. 

Let    D  =  diameter  of  the  driving,  pulley  or  gear 

N  =  number  revolutions  per  minute 

P=force  applied  at  rim 

T  =  twisting  moment. 

The  distance  through  which  P  acts  in  one  minute  is  nDN  in. 
and  work  =  PnDN  in.  Ib.  per  minute. 

PD 
But  —~-  =  T  the  moment,  and  2n:N  =  thQ  angular  velocity. 

.*.  work  =  moment  X  angular  velocity . 
One  horse-power  =  33, 000  ft.  Ib.  per  min. 
=  396,000  in.  Ib.  per  min. 

PxDN  _  27iTN 
'  '  396000  "396000 

TN 
or 


63025 

P  (Q2) 

(93) 


The  general  formula  for  a  circular  shaft  exposed  to  torsion 
alone  is 


But 


where  JV  =  no.  rev.  per  min. 
Substituting  in  formula  for  d 


SN 

167 


168 


MACHINE  DESIGN 


S  may  be  given  the  following  values: 

45,000  for  common  turned  shafting. 

50,000  for  cold  rolled  iron  or  soft  steel. 

65,000  for  machinery  steel. 
It  is  customary  to  use  factors  of  safety  for  shafting  as  follows : 

Headshafts  or  prime  movers 15 

Line  shafting 10 

Short  counters 6 

The  large  factor  of  safety  for  head  shafts  is  used  not  only  on 
account  of  the  severe  service  to  which  such  shafts  are  exposed, 
but  also  on  account  of  the  inconvenience  and  expense  attendant 
on  failure  of  so  important  a  part  of  the  machinery.  The  factor  of 
safety  for  line  shafting  is  supposed  to  be  large  enough  to  allow 
for  the  transverse  stresses  produced  by  weight  of  pulleys,  pull  of 
belts,  etc.,  since  it  is  impracticable  to  calculate  these  accurately 
in  most  cases. 

Substituting  the  values  of  S  and  introducing  factors  of  safety, 
we  have  the  following  formulas  for  the  safe  diameters  of  the 
various  kinds  of  shafts. 

TABLE  XL VI 

DIAMETERS  OF  SHAFTING 


Material 

Kind  of  shaft 

Common  iron 

Soft  steel 

Mach'y  steel 

Head  shaft  

-4? 

«•"  N/f 

4.20  ^/f 

Line  shaft  
Counter  shaft 

3  50  \  HP 

4.00  jJ*P 
3.38  \IHP 

3.67  ^ 
3.10  •*/— 

\  N 

\  N 

\  N 

The  Allis-Chalmers  Co.  base  their  tables  for  the  horse  power 
of  wrought  iron  or  mild  steel  shafting  on  the  formula  HP  =  cd3N 
where  c  has  the  following  values: 


SHAFTING  169 


Heavy  or  main  shafting  .........................  008 

Shaft  carrying  gears  .............................  010 

Light  shafting  with  pulleys  .......................  013 

This  is  equivalent  to  using  values  of  S  as  2570  lb.,  3200  Ib. 
and  4170  lb.  per  square  inch  in  the  respective  classes  —  and  would 
give  for  coefficients  in  Table  XL  VI  the  numbers  5,  4.64  and 
4.25  which  are  somewhat  larger  than  those  given  for  similar  cases 
in  the  table. 

A  table  published  by  Wm.  Sellers  &  Co.  in  their  shafting 
catalogue  —  gives  the  horse-powers  of  iron  and  steel  shafts  for 
given  diameters  and  speeds.  An  investigation  of  the  table 
shows  it  to  be  based  upon  a  value  of  about  4000  Ib.  for  S  or  a 
coefficient  of  4.31  in  Table  XLVI. 

83.  Combined  Torsion  and  Bending.  —  It  frequently  happens 
that  a  shaft  is  subjected  to  bending  as  well  as  torsion;  a  familiar 
example  of  this  is  the  case  of  an  -engine  shaft  which  carries  the 
twisting  moment  due  to  the  crank  effort  and  also  bending  mo- 
ments caused  by  the  overhang  of  the  crank  and  the  weight  of  the 
fly-wheel. 

The  direct  stress  due  to  the  twisting  is  shear  in  the  plane  of 
the  cross-section;  the  stresses  due  to  the  bending  are  primarily 
tension  and  compression  parallel  to  the  axis  and  at  right  angles 
to  the  shearing  stress.  The  combination  of  these  produces  obli- 
que stresses  varying  in  direction  and  intensity  as  the  shaft 
revolves.  It  is  desirable  to  find  the  maximum  values  of  these 
oblique  stresses  whether  shearing  or  tensile. 

Let     p  =  direct  stress  due  to  bending 
q  =  direct  stress  due   to  twisting 
>S;  =  resultant  tensile  stress 
Ss^  resultant  shearing  stress. 

Then  is  it  shown  in  treatises  on  the  mechanics  of  materials 
that  the  maximum  values  of  the  resultant  stresses  are  as  follows:1 


p2  (a) 

1  Merriman's  Mechanics  of  Materials,  p.   151.     Slocum  and  Hancock's 
Strength  of  Materials,  p.  116. 


170  MACHINE  DESIGN 

^  (b) 


2 

Let  M  —  bending  moment  on  shaft 
T  =  twisting  moment  on  shaft. 
Then  by  formulas  (5)  and  (8)  p.  3, 

=  1012M 
_5.177 

Substituting  these  values  in  (a)  and  (b)  and  reducing,  we  have: 

(o) 

(d) 


But  the  bending  moment  which  would  produce  a  stress  =  St  is: 


and  the  twisting  moment  which  would  produce  a  stress  =  S8  is: 


l      5.1 

Combining  these  equations  with   (c)   and  (d)   respectively  and 

reducing: 

(95) 
(96) 

The  method  of  designing  a  shaft  subjected  to  both  bending  and 
twisting  moments  may  thus  be  stated  :  Determine  the  diameter 
of  shaft  necessary  to  withstand  safely  a  bending  moment  Mlt 
Equation  (95)  ;  also,  calculate  the  diameter  to  safely  resist  a 
twisting  moment  7\  (Equation  (96)).  The  larger  diameter 
would  then  be  used,  i.e., 


Equations  (a)  and  (b)  in  this  article  may  be  used  in  combining 
shearing  and  tensile  or  shearing  and  compressive  stresses,  in 
whatever  manner  produced. 


COUPLINGS 


171 


Other  examples  of  combined  stresses  are  furnished  by  columns 
and  by  machine  frames  having  eccentric  loads  (see  Art.  17). 

In  the  case  of  columns,  where  the  load  is  assumed  to  be  central, 
the  empirical  formulas  (12)  and  (12-a)  given  on  pp.  4  and  5 
are  recommended. 

Where  a  material  like  cast  iron  is  concerned,  as  in  the  case  of 
machine  frames,  no  theoretical  analysis  is  of  much  value  and 
reliance  can  be  placed  only  on  experimental  determinations  of 
stresses  and  breaking  loads. 

84.  Couplings.  —  The  flange  or  plate  coupling  is  most  commonly 
used  for  fastening  together  adjacent  lengths  of  shafting. 

Fig.  73  shows  the  proportions 
of  such  a  coupling.  The  flanges 
are  turned  accurately  on  all 
sides,  are  keyed  to  the  shafts 
and  the  two  are  centered  by  the 
projection  of  the  shaft  from  one 
part  into  the  other  as  shown  at 
A.  The  bolts  are  turned  to  fit 
the  holes  loosely  so  as  not  to 
interfere  with  the  alignment. 

The  projecting  rim  as  at  B  pre- 
vents danger  from  belts  catching  on  the  heads  and  nuts  of  the 
bolts. 

The  faces  of  this  coupling  should  be  trued  up  in  a  lathe  after 
being  keyed  to  the  shaft. 

Jones  and  Laughlins  in  their  shafting  catalogue  give  the 
following  proportions  for  flange  couplings. 


FlG 


Diam.  of  shaft 

Diam.  of  hub 

Length  of  hub 

Diam.  of 
coupling 

2 

4* 

3i 

8 

2* 

5f 

4f 

10 

3 

6| 

5* 

12 

3| 

8 

6* 

14 

4 

9 

7 

16 

5 

m 

8! 

20 

172 


MACHINE 'DESIGN 


There  are  five  bolts  in  each  coupling. 

The  sleeve  coupling  is  neater  in  appearance  than  the  flange 
coupling  but  is  more  complicated  and  expensive. 

In  Fig.  74  is  illustrated  a  neat  and  effective  coupling  of  this 
type.  It  consists  of  the  sleeve  S  bored  with  two  tapers  and  two 
threaded  ends  as  shown.  The  two  conical,  split  bushings  BB 


B 


FIG.  74. 


are  prevented  from  turning  by  the  feather  key  K  and  are  forced 
into  the  conical  recesses  by  the  two  threaded  collars  CC  and 
thereby  clamped  firmly  to  the  shaft.  The  key  K  also  nicks 
slightly  the  center  of  the  main  sleeve  S,  thus  locking  the  whole 
combination.  » 

Couplings  similar  to  this  have  been  in  use  in  the  Union  Steel 
Screw  Works,  Cleveland,  Ohio,  for  many  years  and  have  given 
good  satisfaction. 

The  Sellers  coupling  is  of  the 
type  illustrated  in  Fig.  74,  but  is 
tightened  by  three  bolts  running 
parallel  to  the  shaft  and  taking  the 
place  of  the  collars  CC. 

In  another  form  of  sleeve  coup- 
ling the  sleeve  is  split  and  clamped 
to     the     shaft     by    bolts     passing 
through    the   two    halves    as   illus- 
trated in  Fig.  75. 

The  "muff"  coupling,  as  its  name  implies  is  a  plain  sleeve 
slipped  over  the  shafts  at  the  point  of  junction,  accurately 
fitted  and  held  by  a  key  running  from  end  to  end.  It  may  be 
regarded  as  a  permanent  coupling  since  it  is  not  readily  removed. 


(—               "" 

r                  "\ 
i                    j 

i    rAi 

t    rfh    1 

r    iijn 

[    LWJ    ] 

L 

L                    J 

FIG.  75. 


CLUTCHES 


173 


85.  Clutches. — By  the  term  clutch,  is  meant  a  coupling  which 
may  be  readily  disengaged  so  as  to  stop  the  follower  shaft  or 
pulley.  Clutch  couplings  are  of  two  kinds,  positive  or  jaw 
clutches  and  friction  clutches. 

The  jaw  clutch  consists  of  two  hubs  having  sector  shaped 
projections  on  the  adjacent  faces  which  may  interlock.  One 
of  the  couplings  can  be  slid  on  its  shaft  to  and  from  the  other  by 
means  of  a  loose  collar  and  yoke,  so  as  to  engage  or  disengage 
with  its  mate.  This  clutch  has  the  serious  disadvantage  of  not 
being  readily  engaged  when  either  shaft  is  in  motion.  Friction 
clutches  are  not  so  positive  in  action,  but  can  be  engaged  without 
difficulty  and  without  stopping  the  driver. 

Three  different  classes  of  friction  clutches  may  be  distinguished 
according    as    the    engaging 
members  are  flat  rings,  cones 
or  cylinders. 

The  Weston  clutch,  Fig. 
76,  belongs  to  the  first-named 
class.  A  series  of  rings  inside 
a  sleeve  on  the  follower  B  in- 
terlocks with  a  similar  series 
outside  a  smaller  sleeve  on 
the  driver  A  somewhat  as  in 
a  thrust  bearing  (Art.  70). 
Each  ring  can  slide  on  its  sleeve  but  must  rotate  with  it. 

When  the  parts  A  and  B  are  forced  together  the  rings  close  up 
and  engage  by  pairs,  producing  a  considerable  turning  moment 
with  a  moderate  end  pressure.  Let: 

P  =  pressure  along  axis 

n  =  number  of  pairs  of  surfaces  in  contact 

f=  coefficient  of  friction 

r  =  mean  radius  of  ring 

T  =  turning  moment 

Then  will: 

T  =  Pfnr.  (97) 

If  the  rings  are  alternately  wood  and  iron,  as  is  usually  the  case, 
/will  have  values  ranging  from  0.25  to  0.50. 

The  cone  clutch  consists  of  two  conical  frustra,  one  external 


-KB 


FIG.  76. 


174 


MACHINE  DESIGN 


and  one  internal,  engaging  one  another  and  driving  by  friction. 
Using  the  same  notation  as  before,  and  letting  a  =  angle  between 
element  of  cone  and  axis,  the  normal  pressure  between  the  two 

•*  T.I  *•       •        i    *  Ml       1  •*•    J 


surfaces  will  be: 


sin  a. 


and  the  friction  will  be: 


sin  a 


Therefore : 


T=Pfr_ 


sin  a 


(98) 


a  should  slightly  exceed  5  degrees  to  prevent  sticking  and /will 
be  at  least  0.10  for  dry  iron  on  iron. 

Substituting  /=  0.10  and  sin  a  =0.125  we  have  T  =  0.8  Pr  as  a 
convenient  rule  in  designing. 

Fig.  77  illustrates  the  type  of  clutch  more  generally  used  on 
shafting  for  transmitting  moderate  quantities  of  power. 

As  shown  in  the  figure  one 
member  is  attached  to  a  loose 
pulley  on  the  shaft,  but  this 
same  type  can  be  used  for 
connecting  two  independent 
shafts. 

The  ring  or  hoop  H, 
finished  inside  and  out,  is 
gripped  at  intervals  by  pairs 
of  jaws  JJ  having  wooden 
faces. 

These  jaws  are  actuated 
as  shown  by  toggles  and 
levers  connected  with  the 
slip  ring  R.  The  toggles  are 
so  adjusted  as  to  pass  by  the 
center  and  lock  in  the  gripping  position. 

These  clutches  are  convenient  and  durable  but  occupy  con- 
siderable room  in  proportion  to  their  transmitting  power.  The 
Weston  clutch  is  preferable  for  heavy  loads. 

Cork  inserts  in  metal  surfaces  have  been  used  to  some  extent, 
as  the  coefficient  of  friction  is  much  greater  for  cork  than  for 
wood.  The  cork  may  be  boiled  to  soften  it  and  forced  into  holes 
in  one  of  the  members.  When  pressure  is  applied,  the  projecting 
cork  takes  the  load  and  carries  it  with  good  efficiency.  As  the 


FIG.  77. 


CLUTCHES 


175 


normal  pressure  is  increased,  the  cork  yields,  finally  becoming 
flush  with  the  metal  surface  and  dividing  its  load  with  the  latter. 
Cork  in  its  natural  state  is  liable  to  wear  quite  rapidly  under 
hard  service.  It  may  be  hardened  by  being  heated  under  heavy 
pressure  and  in  this  condition  is  much  more  durable. 

Professor  I.  N.  Hollis  gives  the  coefficients  of  friction  for  dif- 
ferent materials  used  in  clutches,  as  follows: 

Cast  iron  on  cast  iron 0.16 

Bronze  on  cast  iron 0.14 

Cork  on  cast  iron 0 . 33 

Professor  C.  M.  Allen  in  experiments  on  clutches  for  looms 
found  that  cork  inserts  gave  a  torque  nearly  double  that  of  a 
leather  face  on  iron. 

The  roller  clutch  is  much  used  on  automatic  machinery  as  it 
combines  the  advantages  of  positive  driving  and  friction  engage- 
ment. A  cylinder  on  the  follower  is  embraced  by  a  rotating 
ring  carried  by  the  driver. 

The  ring  has  a  number  of  recesses  on  its  inner  surface  which 
hold  hardened  steel  rollers.  These  recesses  being  deeper  at  one 
end  allow  the  rollers  to  turn  freely  as  long  as  they  remain  in  the 
deep  portions. 

The  bottom  of  the  recess  is  inclined  to  the  tangent  of  the  circle 
at  an  angle  of  from  9  to  14  degrees. 

When  by  suitable  mechanism  the  rollers  are  shifted  to  the 
shallow  portions  of  the  recesses  they  are  immediately  gripped 
between  the  ring  and  the  cylinder  and  set  the  latter  in  motion. 

A  clutch  of  this  type  is  almost  instantaneous  in  its  action  and 
is  very  powerful,  being  limited  only  by  the  strength  of  the 
materials  of  which  it  is  composed. 

Several  small  rolls  of  different  materials  and  diameters  were 
tested  by  the  writer  in  1905  with  the  following  results: 


Material 

Diameter 

Length 

Set  load 

Ultimate  load 

Cast  iron         

0.375 

1.5 

5,500 

12,400 

Cast  iron 

0  75 

1.5 

6,800 

19,500 

Cast  iron  
Cast  iron 

1.125 
0.4375 

1.5 
1.5 

7,800 
8,800 

29,700 
20,000 

Soft  steel  

0.4375 

1.5 

11,100 

Hard  steel       .... 

0.4375 

1.5 

35,000 

176  MACHINE  DESIGN 

86.  Automobile  Clutches. — The  development  of  the  automobile 
industry  has  created  a  demand  for  clutches  of  small  size  and 
considerable  power;  these  clutches  must  also  be  capable  of  picking 
up  a  load  gently  and  of  holding  it  firmly;  they  must  be  durable 
and  reliable  under  peculiarly  severe  conditions  and  for  consider- 
able periods. 

Mr.  Henry  Souther  contributes  to  the  literature  of  this  subject 
an  interesting  paper  from  which  some  of  the  following  data  are 
quoted.  Reference  is  made  to  the  paper  itself  for  more  complete 
information.1 

Automobile  clutches  may  be  roughly  classified  as  (a)  conical; 
(b)  disc  or  multiple  disc;  (c)  band  either  expanding  or  contracting. 
The  clutch  is  located  between  the  engine  and  gear  box,  usually 
near  the  fly-wheel  and  sometimes  forming  a  part  of  it. 

Conical  clutches  are  in  some  respects  the  most  satisfactory  for 
automobile  use.  They  require  but  slight  motion  for  engagement 
and  slight  pressure  to  hold  them  in  place.  No  lubricant  is 
necessary  and  therefore  there  is  no  trouble  from  gumming  and 
sticking. 

The  materials  used  for  the  rubbing  surfaces  are  generally 
aluminum  covered  with  leather  for  one,  and  gray  cast  iron  for  the 
other.  Castor  or  neatsfoot  oil  may  be  used  to  keep  the  leather 
soft.  To  render  the  engagement  more  gradual,  springs  are 
sometimes  placed  under  the  leather  at  six  or  eight  points  on  the 
circumference;  these  permit  some  slipping  until  the  whole 
surface  of  the  leather  is  brought  into  contact. 

The  angle  of  the  cone  is  about  8  degrees  in  ordinary  practice, 
but  some  manufacturers  are  using  10  or  12  degrees.  (This  is 
the  angle  on  one  side.) 

The  principal  difficulty  with  conical  clutches  is  that  of  poor 
alignment.  Unless  the  axes  of  the  two  cones  coincide,  engage- 
ment is  uncertain  and  irregular.  This  coincidence  can  only  be 
secured  by  the  use  of  two  universal  joints  insuring  perfect 
flexibility. 

Mr.  Souther  gives  the  following  table  as  representing  three 
typical  clutches  in  successful  use: 

1  Trans.  A.  S.  M.  E.,  May,  1908. 


CLUTCHES 


177 


TABLE  XL VII 

POWER  OF  CLUTCHES 


1 

2 

3 

Area  of  surface  (square  inches) 

113  1 

78  7 

73  6 

Angle  (one  side)  (degrees)  
Maximum  radius  (inches)           .    . 

8 
8£ 

8 
8£ 

8 
7£ 

Spring  pressure  (pounds)  

375 

320 

250 

Horse-power                        

48 

42 

40 

Fig.   78  illustrates  the  conical  clutch  in  its  simplest  form. 

The  disc  dutch  consists  of  a  disc  on  the  driven  member  clamped 
between  two  discs  on  the  driver,  which  latter  is  generally  the 
fly-wheel.  Springs  are  used  to  insure  separation  when  dis- 
engaged and  other  springs  furnish  the  pressure  for  engagement. 

A  multiple  disc  clutch  similar  to  the  Weston  is  also  used.  In 
this  case  the  discs  are  alter- 
nately of  bronze  and  steel. 
All  disc  clutches  must  be 
lubricated  and  upon  the 
type  and  quantity  of  lubri- 
cation depends  the  character 
of  the  service.  Copious 
lubrication  means  gradual 
engagement  and  slight  driv- 
ing power;  scanty  lubrica- 
tion gives  more  power  and 
quick  seizure. 

The  principal  disadvan- 
tage of  the  disc  clutch  is 
the  heavy  spring  pressure 
necessary  to  insure  driving 
power. 

The  multiple  disc  clutches  cause  some  trouble  in  lubrication 
and  are  complicated  and  difficult  of  access. 

Band  clutches  depend  for  their  driving  power  on  the  friction 
between  the  case  and  an  adjustable"  band  or  ring  which  can  be 
expanded  or  contracted  by  suitable  mechanism. 


FIG.  78. 


178  MACHINE  DESIGN 

The  more  usual  construction  has  a  band  which  is  expanded 
against  the  inside  surface  of  the  enclosing  case  by  means  of 
internally  operated  levers  and  springs. 

Centrifugal  force  at  high  speeds  has  a  disturbing  effect  on 
the  levers  and  sometimes  causes  the  clutch  to  release  auto- 
matically. This  difficulty  has  been  overcome  in  some  clutches 
by  an  improved  arrangement  of  levers  and  springs. 

87.  Coupling  Bolts.  —  The  bolts  used  in  the  ordinary  flange 
couplings  are  exposed  to  shearing,  and  the  combined  moment  of 
the  shearing  forces  should  equal  the  twisting  moment  on  the 
shaft. 

Let  n  =  number  of  bolts 
d1  =  diameter  of  bolt 
D  =  diameter  of  bolt  circle. 

We  will  assume  that  the  bolt  has  the  same  shearing  strength 
as  the  shaft.     The  combined  shearing  strength  of  the  bolts  is 
and  their  moment  of  resistance  to  shearing  is 


This  last  should  equal  the  torsion  moment  of  the  shaft  or 


o.l 

Solving  for  dl  and  assuming  D=3d  as  an  average  value,  we 
have  ^==,  (79) 


In  practice  rather  larger  values  are  used  than  would  be  given 
by  the  formula. 

88.  Shafting  Keys.  —  The  moment  of  the  shearing  stress  on  a 
key  must  also  equal  the  twisting  moment  of  the  shaft. 

Let  6  =  breadth  of  a  key 
1  =  length  of  key 
h  =  total  depth  of  key 
S'  =  shearing  strength  of  key. 


SHAFTING  KEYS  179 

The  moment  of  shearing  stress  on  key  is 


and  this  must  equal  -^r  Usually  b  =  ~.- 

O.I  4 

For  shafts  of  machine  steel  S  =  S',  and  for  iron  shafts  S  =  %S' 
nearly,  as  keys  should  always  be  of  steel. 
Substituting  these  values  and  reducing: 
For  iron  shafting  l  =  1.2d  nearly. 

For  steel  shafting  l  =  1.6d  nearly  as  the  least  lengths 

of  key  to  prevent  its  failing  by  shear. 

If  the  keyway  is  to  be  designed  for  uniform  strength,  the  shear- 
ing area  of  the  shaft  on  the  line  AB,  Fig.  79,  should  equla  the 
shearing  area  of  the  key,  if  shaft  and  key 
are  of  the  same  material  and  AB  =  CD  =  b. 

These  proportions  will  make  the  depth 
of  keyway  in  shaft  about  =ffr  and  would 
be  appropriate  for  a  square  key. 

To  avoid  such  a  depth  of  keyway  which 
might  weaken  the  shaft,  it  is  better  to  use 
keys  longer  than  required  by  preceding  for- 
mulas. In  American  practice  the  total 
depth  of  key  rarely  exceeds  J6  and  one-half 
of  this  depth  is  in  shaft. 

To  prevent  crushing  of  the  key  the  moment  of  the  compressive 
strength  of  half  the  depth  of  key  must  equal  T. 

dlh^         Sd* 
or  2X^X    c  =  lU  (a) 

where  Sc  is  the  compressive  strength  of  the  key. 
For  iron  shafts  SC  =  2S 


and  for  steel  shafts  Sc  =  =  S 


Substituting  values  of  Sc  and  assuming  h  =  %b  =  -fad  we  have 
Iron  shafts  l  =  2.5d  nearly. 

Steel  shafts  l  =  3^d  nearly,  as  the  least  length  for 

flat  keys  to  prevent  lateral  crushing. 


180 


MACHINE  DESIGN 


The  above  refers  to  parallel  keys.  Taper  keys  have  parallel 
sides,  but  taper  slightly  between  top  and  bottom.  When 
driven  home  they  have  a  tendency  to  tip  the  wheel  or  coupling 
on  the  shaft.  This  may  be  partially  obviated  by  using  two  keys 
90  degrees  apart  so  as  to  give  three  points  of  contact  between 

hub  and  shaft.     The  taper  of  the  keys 
is  usually  about  |  in.  to  1  ft. 

The  Woodruff  key  is  sometimes 
used  on  shafting.  As  may  be  seen  in 
Fig.  80  this  key  is  semi-circular  in 
shape  and  fits  a  recess  sunk  in  the 
shaft  by  a  milling  cutter. 

89.  Strength  of  Keyed  Shafts.— Some 
very  interesting  experiments  on  the 
strength  of  shafts  with  keyways  are 
reported  by  Professor  H.  F.  Mo^re  * 
The  material  of  the  shafts  was  "soft 
steel  some  being  turned  and  some 
cold-rolled.  The  diameters  varied  from 
1£  to  2\  in.  Keyways  of  ordinary  pro- 
portions, both  for  straight  keys  and  for  Woodruff  keys,  were  cut 
in  the  specimens  and  the  latter  were  then  subjected  to  twisting 
and  to  combined  twisting  and  bending. 

So  far  as  the  ultimate  strength  was  concerned,  the  keyways 
seemed  to  have  little  effect,  the  shaft  with  a  single  keyway 
having  about  the  same  strength  as  a  shaft  without  the  keyway. 
After  the  elastic  limit  was  passed,  the  keyways  gradually  closed 
up  and  were  entirely  closed  at  rupture.  The  elastic  limit, 
however,  was  noticeably  affected  by  the  presence  of  a  keyway. 
The  ratio  of  the  strength  at  elastic  limit  with  keyway  to  the 
strength  at  elastic  limit  without  keyway  is  called  the  efficiency 
and  is  denoted  by  -e-.  The  corresponding  ratio  of  angles  of 
twist  inside  the  elastic  limit  is  denominated  —k—. 

According  to  Professor  Moore,  the  following  equations  repre- 
sent fairly  well  the  values  of  e  and  k : 

e  =  l-Q.2w-l.lh  (99) 

h  (100) 


FIG.   80. 


University  of  Illinois  Bulletin  No.  42,  1909. 


SHAFTING  KEYS 


181 


where  w  = 


and  h  = 


width  of  keyway 
diameter  of  shaft 
depth  of  keyway 


diameter  of  shaft 
Two  values  of  w  were  used  in  the  experiments:  w  =  Q.25  and  0.50 
and  two  values  of  h: 

h  =  Q.125  and  0.1875. 

Table  XLVIII  gives  the  values  of  e  as  obtained  by  the  experi- 
ments: 

TABLE  XLVIII 

EFFICIENCY  OF  SHAFTS  WIFH  KEYWAYS 

^  „  .  elastic  strength  of  shaft  with  keyway 

Efficiency  =  -r  — r—  ,  ,    . — r-r       -^ — 

elastic  strength  of  shaft  without  keyway 


Dimensions  of  keyway 

17  =  0.50 
A  =  0.125 

W  =  0.25 
h  =  0.1875 

PF  =  0.25 
A  =  0.125 

Woodruff 
System1 

Under  simple  torsion: 
Cold-rolled  shaft,  diameter, 
11  in. 
Cold-rolled    shaft,    diameter, 
1  9/16  in. 

0.762 

0.803 
0.758 

0.760 

0.846 
0.817 

0.820 

0.900 
0.889 

0.840 

0.860 
0.815 

Cold-rolled     shaft,     diameter, 
1  15/16  in. 

0.748 
0.764 

0.710 
0.750 

0.860 
0.824 

0.826 
0.835 

Cold-rolled     shaft,     diameter, 
21  in. 

0.848 
0.705 

0.775 
0.689 

0.839 
0.825 

0.943 
0.861 

Under  combined   torsion   and 
bending: 
1.  Twisting    moment  =  bend- 
ing moment. 
Cold-rolled    shaft,    diameter, 
11  in. 

0.630 
0.680 

0.636 
0.698 

0.791 
0.803 

0.716 
0.750 

Cold-rolled     shaft,     diameter, 
1  15/16  in 

0.584 
0.671 

0.697 
0  775 

0.854 

0.858 
0  840 

2.  Twisting  moment  =  5/3 
bending  moment. 
Cold-rolled     shaft,     diameter, 
11  in. 

0.895 
0.870 

0.670 
0.735 

0.940 

0.888 

0.930" 
0.880 

Cold-rolled     shaft,     diameter, 
1  15/16  in. 

0.740 
0.815 

0.832 
0.840 

0.856 
0.810 

General  average  

0.752 

0.735 

0.850 

0.845 

lln  1  1/4-in.  shafts  keyways  were  cut  for  No.  15  Woodruff  keys. 
In  1  9/16-in.  shafts  keyways  were  cut  for  No.  25  Woodruff  keys. 
In  1  15/16-in.  shafts  keyways  were  cut  for  No.  S  Woodruff  keys. 
In  2  1/4-in.  shafts  keyways  were  cut  for  No.  U  Woodruff  keys. 


182 


MACHINE  DESIGN 


The  average  value  of  the  fiber  stress  of  the  cold-rolled  shafting 
at  the  elastic  limit  was  38940  Ib.  and  the  average  modulus  of 
elasticity  11,985,000. 

It  would  appear  that,  considering  the  factor  of  safety  usually 
allowed  in  shafting,  the  effect  of  ordinary  keyways  can  safely  be 
neglected. 

90.  Hangers  and  Boxes. — Since  shafting  is  usually  hung  to  the 
ceiling  and  walls  of  buildings  it  is  necessary  to  provide  means 


FIG.  83. 


FIG.  84. 


for  adjusting  and  aligning  the  bearings  as  the  movement  of  the 
building  disturbs  them.  Furthermore  as  line  shafting  is  contin- 
uous and  is  not  perfectly  true  and  straight,  the  bearings  should 
be  to  a  certain  extent  self-adjusting.  Reliable  experiments 


HANGERS 


183 


have  shown  that  usually  one-half  of  the  power  developed  by  an 
engine  is  lost  in  the  friction  of  shafting  and  belts.  It  is  important 
that  this  loss  be  prevented  as  far  as  possible. 

The  boxes  are  in  two  parts  and  may  be  of  bored  cast-iron  or 
lined  with  Babbitt  metal.  They  are  usually  about  four  diam- 
eters of  the  shaft  in  length  and  are  oiled  by  means  of  a  well  and 
rings  or  wicks.  (See  Art.  58.) 
The  best  method  of  support- 
ing the  box  in  the  hanger  is 
by  the  ball-and-socket  joint; 
all  other  contrivances  such  as 
set  screws  are  but  poor  sub-  FIG.  85. 

stitutes. 

Fig.  81  shows  the  usual  arrangement  of  the  ball  and  socket. 

A  and  B  are  the  two  parts  of  the  box.     The  center,  is  cast  in 

the  shape  of  a  partial  sphere  with  C  as  a  center  as  shown  by  the 

dotted  lines.     The  two  sockets  S  S  can  be  adjusted  vertically  in 

the  hanger  by  means  of  screws  and  lock  nuts.     The  horizontal 

adjustment  of  the  hang- 
er is  usually  effected  by 
moving  it  bodily  on  the 
support,  the  bolt  holes 
being  slotted  for  this 
purpose. 

Counter  shafts  are 
short  and  light  and  are 
not  subject  to  much 
bending.  Consequently 
there  is  not  the  same 


need  of  adjustment  as  in 
FlG-  86-  line  shafting. 

In  Fig.  82  is  illustrated  a  simple  bearing  for  counters.  The 
solid  cast-iron  box  B  with  a  spherical  center  is  fitted  directly  in  a 
socket  in  the  hanger  H  and  held  in  position  by  the  cap  C  and  a  set 
screw.  There  is  not  space  here  to  show  all  the  various  forms 
of  hangers  and  floor  stands  and  reference  is  made  to  the  catalogues 
of  manufacturers.  Hangers  should  be  symmetrical,  i.e.,  the 
center  of  the  box  should  be  in  a  vertical  line  with  center  of  base. 
They  should  have  relatively  broad  bases  and  should  have  the 


184  MACHINE  DESIGN 

metal  disposed  to  secure  the  greatest  rigidity  possible.  Cored 
sections  are  to  be  preferred, 

Fig.  83  illustrates  the  proportions  of  a  Sellers  line-shaft  hanger. 
This  type  is  also  made  with  the  lower  half  removable  so  as  to 
facilitate  taking  down  the  shaft. 

Fig.  84  shows  the  outlines  of  a  hanger  for  heavy  shafting  as 
manufactured  by  the  Jones  &  Laughlins  Company  while  Fig.  85 
illustrates  the  design  of  the  box  with  oil  wells  and  rings. 

The  open  side  hanger  is  sometimes  adopted  on  account  of  the 
ease  with  which  the  shaft  can  be  removed,  but  it  is  much  less 
rigid  than  the  closed  hanger  and  is  suitable  only  for  light  shafting. 
The  countershaft  hanger  shown  in  Fig.  86  is  simple,  strong  and 
symmetrical  and  is  a  great  improvement  over  those  using  pointed 
set  screws  for  pivots.  Hangers  similar  to  this  are  used  by  the 
Brown  &  Sharpe  Mfg.  Co.  with  some  of  their  machines. 

PROBLEMS 

1.  Calculate  the  safe  diameters  of  head  shaft  and  three  line  shafts  for  a 
factory,  the  material  to  be  rolled  iron  and  the  speeds  and  horse-powers  as 
follows : 

Head  shaft  100  H.  P.  200  rev.  per  min. 

Machine  shop  30  H.  P.  120  rev.  per  min. 

Pattern  shop  50  H.  P.  250  rev.  per  min. 

Forge  shop  20  H.  P.  200  rev.  per  min. 

2.  Determine  the  horse-power  of  at  least  two  lines  of  shafting  whose 
speeds  and  diameters  are  known. 

3.  Design  and  sketch  to  scale  a  flange  coupling  for  a  3-in.  line  shaft 
including  bolts  and  keys. 

4.  Design  a  sleeve  coupling  for  the  foregoing,  different  in  principle  from 
the  ones  shown  in  the  text. 

5.  A  4-in.  steel  head  shaft  makes  100  rev.  per  min.     Find  the  horse-power 
which  it  will  safely  transmit,  and  design  a  Weston  ring  clutch  capable  of 
carrying  the  load. 

There  are  to  be  six  wooden  rings  and  five  iron  rings  of  12-in.  mean  diame- 
ter. Find  the  moment  carried  by  each  pair  of  surfaces  in  contact  and  the 
end  pressure  required. 

6.  Find  mean  diameter  of  a  single  cone  clutch  for  same  shaft  with  same 
end  pressure. 

7.  Find  radial  pressure  required  for  a  clutch  like  that  shown  in  Fig.  77,  the 
ring  being  24  in.  in  mean  diameter  and  there  being  four  pairs  of  grips.     Other 
conditions  as  in  preceding  problems. 


HANGERS  185 

8.  Select  the  line-shaft  hanger   which  you   prefer  among  ^hose  in  the 
laboratories  and  make  sketch  and  description  of  the  same. 

9.  Do.  for  a  countershaft  hanger. 

10.  Explain  in  what  way  a  floor-stand  differs  from  a  hanger. 

REFERENCES 

Machine  Design.     Low  and  Be  vis,  Chapter  VIII. 

Efficiency  of  Shafting.     Tr.  A.  S.  M.  E.,  Vol.  VI,  p.  461;  Vol.  VII,  p.  138; 

Vol.  XVIII,  p.  228;  Vol.  XVIII,  p.  861. 
Shafting  Clutches.     Tr.  A.  S.  M.  E.,  Vol.  XIII,  p.  236. 
Ball  Bearing  Hangers.     Tr.  A.  S.  M.  E.,  Vol.  XXXII,  p.  533. 
Test  of  Clutch  Coupling.     Tr.  A.  S.  M.  E.,  Vol.  XXXII,  p.  549. 


CHAPTER  X 
GEARS,  PULLEYS  AND  CRANKS 

91.  Gear  Teeth. — The  teeth  of  gears  may  be  either  cast  or  cut, 
but  the  latter  method  prevails,  since  cut  gears  are  more  accurate 
and  run  more  smoothly  and  quietly.  The  proportions  of  the 
teeth  are  essentially  the  same  for  the  two  classes,  save  that  more 
back  lash  must  be  allowed  for  the  cast  teeth.  The  circular 
pitch  is  obtained  by  dividing  the  circumference  of  the  pitch 
circle  by  the  number  of  teeth.  The  diametral  pitch  is  obtained 
by  dividing  the  number  of  teeth  by  the  diameter  of  the  pitch 
circle  and  equals  the  number  of  teeth  per  inch  of  diameter. 
The  reciprocal  of  the  diametral  pitch  is  sometimes  called  the 
module.  The  addendum  is  the  radial  projection  of  the  tooth 
beyond  the  pitch  circle,  the  dedendum  the  corresponding- 
distance  inside  the  pitch  circle.  The  clearance  is  the  difference 
between  the  dedendum  and  addendum;  the  back  lash  the  differ- 
ence between  the  widths  of  space  and  tooth  on  the  pitch  circle. 

Let  circular  pitch  =p 

module  =-=m 

71 

diametral  pitch  =  -=— 
p     m 

addendum  =a 

dedendum  or  flank  =f 

clearance  =f—a  =  c 

height  =a+f=h 

width  =w.  (See  Fig.  88.) 

The  usual  rule  for  standard  cut  teeth  is  to  make  w  =  ^,  allowing 

2i 

gm 
no  calculable  back-lash,  to  make  a  =  m  and  f=-^-  or  h  =  2^m 

o 

and  clearance  =  •=-' 

o 

There  is,  however,  a  marked  tendency  at  the  present  J^ime 
toward  the  use  of  shorter  teeth.  The  reasons  urged  for  their 

186 


GEAR  TEETH 


187 


adoption  are:  first,  greater  strength  and  less  obliquity  of  action; 
second,  less  expense  in  cutting.1  Several  systems  have  been 
proposed  in  which  the  height  of  tooth  h  varies  from  0.425p  to 


According  to  the  latter  system  a  —  Q.25p,f=Q.3p,  and  c  =  .05p. 

In  modern  practice  the  diametral  pitch  is  a  whole  number  or  a 
common  fraction  and  is  used  in  describing  the  gear.  For 
instance,  a  3-pitch  gear  is  one  having  3  teeth  per  inch  of  diameter. 
The  following  table  gives  the  pitches  in  common  use  and  the 
proportions  of  long  and  short  teeth. 

7) 

If  the  gears  are  cut,  w  —  ^>  if  cast  gears  are  used,  w  —  0.46p 
to  QASp. 

TABLE  XLIX 
PROPORTIONS  OP  GEAR  TEETH 


Pitch 

Standard  teeth 

Short  teeth 

Diametral 

Circular 

Addend,  a 

Height  h 

Clear- 

Addend,  a 

Height  h 

Clear- 

ance c 

ance  c 

i 

6.283 

2. 

4.25 

0.25 

1.571 

3.456 

0.314 

1 

4.189 

1.33 

2.82 

0.167 

1.047 

2.303 

0.209 

1 

3.142 

1. 

2.125 

0.125 

0.785 

1.728 

0.157 

H 

2.513 

0.8 

1.7 

0.1 

0.628 

1.383 

0.125 

li 

2.094 

0.667 

1.415 

0.083 

0.524 

1.152 

0.105 

H 

1.795 

0.571 

1.212 

0.071 

0.449 

0.988 

0.09 

2 

1.571 

0.5 

1.062 

0.062 

0.392 

0.863 

0.078 

21 

1.396 

0.445 

0.945 

0.056 

0.349 

0.768 

0.070 

2i 

1.257 

0.4 

0.85 

0.05 

0.314 

0.691 

0.063 

2f 

1.142 

0.364 

0.775 

0.045 

0.286 

0.629 

0.057 

3 

1.047 

0.333 

0.708 

0.042 

0.262 

0.576 

0.052 

3J 

0.898 

0.286 

0.608 

0.036 

0.224 

0.494 

0.045 

4 

0.785 

0.25 

0.531 

0.031 

0.196 

0.432 

0.039 

5 

0.628 

0.2 

0.425 

0.025 

0.157 

0.345 

0.031 

6 

0.524 

0.167 

0.354 

0.021 

0.131 

0.288 

0.026 

7 

0.449 

0.143 

0.304 

0.018 

0.112 

0.246 

0.022 

8 

0.393 

0.125 

0.266 

0.016 

0.098 

0.216 

0.020 

9 

0.349 

0.111 

0.236 

0.014 

0.087 

0.191 

0.017 

10 

0.314 

0.1 

0.212 

0.012 

0.079 

0.174 

0.016 

11 

0.286 

0.091 

0.193 

0.011 

0.071 

0.156 

0.014 

12 

0.262 

0.0834 

0.177 

0.010 

0.065 

0.143 

0.013 

13 

0.242 

0.077 

0.164 

0.010 

0.060 

0.132 

0.012 

14 

0.224 

0.0715 

0.152 

0.009 

0.056 

0.123 

0.011 

15 

0.209 

0.0667 

0.142 

0.008 

0.052 

0.114 

0.010 

16 

0.196 

0.0625 

0.133 

0.008 

0.049 

0.108 

0.010 

1  See  Am.  Mach.  Jan.  7,  1897,  p.  6. 


188 


MACHINE  DESIGN 


92.  Strength  of  Teeth. — Let  P  =  total  driving  pressure  on 
wheel  at  pitch  circle.  This  may  be  distributed  over  two  or  more 
teeth,  but  the  chances  are  against  an  even  distribution. 

Again,  in  designing  a  set  of  gears  the  contact  is  likely  to  be 
confined  to  one  pair  of  teeth  in  the  smaller  pinions. 

Each  tooth  should  therefore  be  made  strong  enough  to  sustain 
the  whole  pressure. 

Rough  Teeth. — The  teeth  of  pattern  molded  gears  are  apt 
to  be  more  or  less  irregular  in  shape,  and  are  especially  liable 
to  be  thicker  at  one  end  on  account  of  the  draft  of  the 
pattern. 

In  this  case  the  entire  pressure  may  come  on  the  outer  corner 
of  a  tooth  and  tend  to  cause  a  diagonal  fracture. 

Let  C  in  Fig.  87  be  the  point  of  application  of  the  pressure  P, 
and  A B  the  line  of  probable  fracture. 


B 

Drop  the 
_LCD  on 
AB 

Let  AB  =  x 
A              and 
CD=y 
angle 
CAD  =  a 

\~7 

1     X»- 

i         \ 

WM^:       '  'WS> 

FIG.  87. 

The  bending  moment  at  section  AB  is  M  =  Pfy,  aiid  the  moment 
of  resistance  is  M'  =  ^Sxw2 

where  $  =  safe  transverse  strength 
of  material. 


and 


QPy 

W2X 


(a) 


If  P  and  w  are  constant,  then  S  is  a  maximum  when  -  is   a 
maximum. 


GEAR  TEETH  189 

But  y  =  h  sina  and  x  =  — 

cos  a 

-  —sina  cosa  which  is  a  maximum 
x 

when   a  -=45°  and  ^  =  1 
x 

3P 

Substituting  this  value  in  (a)  we  have  S  =  —  ^ 

3  P 

But  in  this  case  w  =  A7p  and  therefore  S=  2 

and  p  =  3.684  \^  (101) 

1  fs" 

diametral  pitch,        -  =  .853  \^.  (102) 

//&•  x^ 

Unless  machine  molded  teeth  are  very  carefully  made,  it  may 
be  necessary  to  apply  this  rule  to  them  as  well. 

Cut  Gears.  —  With  careful  workmanship  machine  molded  and 
machine  cut  teeth  should  touch  along  the  whole  breadth.  In 
such  cases  we  may  assume  a  line  of  contact  at  crest  of  tooth  and  a 
maximum  bending  moment. 

M  =  Ph. 

The  moment  of  resistance  at  base  of  tooth  is 


when  6  is  the  breadth  of  tooth. 

In  most  teeth  the  thickness  at  base  is  greater  than  w,  but  in 
radial  teeth  it  is  less.     Assuming  standard  proportions  for  cut 


w  =  .5p 
and  substituting  above: 

.6765  Pp= 


P  =  .06166Sp.  (103) 

For  short  teeth  having  h  =  .55p  formula   (103)    reduces  to: 
P  =  .075SbSp.  (104) 

The  above  formulas  are  general  whatever  the  ratio  of  breadth 


190  MACHINE  DESIGN 

to   pitch.     The  general   practice   in  this   country  is  to   make 

b=3p. 
Substituting  this  value  of  b  in  (103)  and  (104)  and  reducing: 

Long  teeth :  p  =  2.326\^  (105) 

Short  teeth:  p=  2.098 \L-  (106) 

The    corresponding   formulas   for   the    diametral   pitch    are: 

1  l~Cf 

Long  teeth:  -  =  1.35  \~  (107) 

Short  teeth:  ~  =  1.49  \^  (108) 

93.  Lewis'  Formulas. — The  foregoing  formulas  can  only  be 
regarded  as  approximate,  since  the  strength  of  gear  teeth  depends 
upon  the  number  of  teeth  in  the  wheel;  the  teeth  of  a  rack  are 
broader  at  the  base  and  consequently  stronger  than  those  of  a 
pinion.  This  is  more  particularly  true  of  epicycloidal  teeth. 
Mr.  Wilfred  Lewis  has  deduced  formulas  which  take  into  account 
this  variation.  For  cut  spur  gears  of  standard  dimensions  the 
Lewis  formula  is  as  follows: 


(109) 

where  n  =  number  of  teeth. 

This  formula  reduces  to  the  same  as  (103),  for  n  =  14  nearly. 

Formula  (103)  would  then  properly  apply  only  to  small  pin- 
ions, but  as  it  would  err  on  the  safe  side  for  larger  wheels,  it  can 
be  used  where  great  accuracy  is  not  needed.  The  same  criticism 
applies  to  the  other  formulas  in  Art.  92. 

The  value  of  S  used  should  depend  on  the  material  and  on  the 
speed. 

The  following  safe  values  are  recommended  for  cast  iron  and 
cast  steel. 


GEAR  TEETH 


191 


Linear  velocity 
ft.  per  min. 

100 

200 

300 

600 

900 

1200 

• 
1800 

2400 

Cast  iron 

8000 

6  000 

4,800 

4,000 

3,000 

2,400 

2,000 

1  700 

Cast  steel  

24,000 

15,000 

12,000 

10,000 

7,500 

6,000 

5,000 

4,250 

For  gears  used  in  hoisting  machinery  where  there  is  slow  speed 
and  liability  of  shocks  a  writer  in  the  Am.  Mach.  recommends 
smaller  values  of  S  than  those  given  above1  and  proposes  the 
following  for  four  different  metals: 


Linear  velocity 
ft.  per  min. 

100 

200 

300 

600 

900 

1200 

1800 

2400 

Gray  iron  

4,800 

4,200 

3,840 

3,200 

2,400 

1,920 

1,600 

1,360 

Gun  metal  

7,200 

6,300 

5,760 

4,800 

3,600 

2,880 

2,400 

2,040 

Cast  steel  

9,600 

8,400 

7,680 

6,400 

4,800 

3,840 

3,200 

2,720 

Mild  steel  

12,000 

10,500 

9,600 

8,000 

6,000 

4,800 

4,000 

3,400 

The  experiments  described  in  the  next  article  show  that  the 
ultimate  values  of  S  are  much  less  than  the  transverse  strength 
of  the  material  and  point  to  the  need  of  large  factors  of  safety. 

94.  Experimental  Data.  —  In  the  Am.  Mach.  for  Jan.  14,  1897, 
are  given  the  actual  breaking  loads  of  gear  teeth  which  failed  in 
service.  The  teeth  had  an  average  pitch  of  about  5  in.,  a  breadth 
of  about  18  in.  and  the  rather  unusual  velocity  of  over  2000  ft. 
per  minute.  The  average  breaking  load  was  about  15,000  Ib. 
there  being  an  average  of  about  50  teeth  on  the  pinions.  Substi- 
tuting these  values  in  (109)  and  solving  we  get 

£  =  1575  Ib. 

This  very  low  value  is  to  be  attributed  to  the  condition  of 
pressure  on  one  corner  noted  in  Art.  92.  Substituting  in  formula 
for  such  a  case. 

op 


This  all  goes  to  show  that  it  is  well  to  allow  large  factors  of 
safety  for  rough  gears,  especially  when  the  speed  is  high. 
1  Am.  Mach.,  Feb.  16,  1905. 


192  MACHINE  DESIGN 

Experiments  have  been  made  by  the  author  on  the  static 
strength  of  rough  cast-iron  gear  teeth  by  breaking  them  in  a  test- 
ing machine.  The  teeth  were  cast  singly  from  patterns,  were 
two  pitch  and  about  6  in.  broad.  The  patterns  were  constructed 
accurately  from  templates  representing  15  degrees  involute 
teeth  and  cycloidal  teeth  drawn  with  a  describing  circle  one-half 
the  pitch  circle  of  15  teeth;  the  proportions  used  were  those  given 
for  standard  cut  gears. 

There  were  in  all  41  cycloidal  teeth  of  shapes  corresponding  to 
wheels  of  15-24-36-48-72-120  teeth  and  a  rack.  There  were 
28  involute  teeth  corresponding  to  numbers  above  given  omitting 
the  pinion  of  15  teeth. 

The  pressure  was  applied  by  a  steel  plunger  tangent  to  the 
surface  of  tooth  and  so  pivoted  as  to  bear  evenly  across  the  whole 
breadth.  The  teeth  were  inclined  at  various  angles  so  as  to 
vary  the  obliquity  from  0  to  25  degrees  for  the  cycloidal  and  from 
15  degrees  to  25  degrees  for  the  involute.  The  point  of  applica- 
tion changed  accordingly  from  the  pitch  line  to  the  crest  of  the 
tooth.  From  these  experiments  the  following  conclusions  are 
drawn : 

1.  The  plane  of  fracture  is  approximately  parallel  to  line  of 
pressure  and  not  necessarily  at  right  angles  to  radial  line  through 
center  of  tooth. 

2.  Corner  breaks  are  likely  to  occur  even  when  the  pressure  is 
apparently  uniform  along  the  tooth.     There  were  fourteen  such 
breaks  in  all. 

3.  With  teeth  of  dimensions  given,  the  breaking  pressure  per 
tooth  varies  from  25,000  Ib.  to  50,000  Ib.  for  cycloids  as  the  num- 
'ber  of  teeth  increases  from  15  to  infinity;  the  breaking  pressure 
for  involutes  of  the  same  pitch  varies  from  34,000  Ib.  to  80,000  Ib. 
as  the  number  increases  from  24  to  infinity. 

4.  With  teeth  as  above  the  average  breaking  pressure  varies 
from  50,000  Ib.  to  26,000  Ib.  in  the  cycloids  as  the  angle  changes 
from  0  degrees  to  25  degrees  and  the  tangent  point  moves  from 
pitch  line  to  crest;  with  involute  teeth  the  range  is  betwreen  64,000 
and  39,000  Ib. 

5.  Reasoning  from  the  figures  just  given,  rack  teeth  are  about 
twice  as  strong  as  pinion  teeth  and  involute  teeth  have  an  advan- 
tage in  strength  over  cycloidal  of  from  40  to  50  per  cent.     The 


GEAR  TEETH  193 

advantage  of  short  teeth  in  point  of  strength  can  also  be  seen. 
The  modulus  of  rupture  of  the  material  used  was  about  36,000  Ib. 
Values  of  S  calculated  from  Lewis'  formula  for  the  various  tooth 
numbers  are  quite  uniform  and  average  about  40,000  Ib.  for 
cycloidal  teeth.  Involute  teeth  are  to-day  generally  preferred 
by  manufacturers. 

95.  Modern  Practice. — Two  tendencies  are  quite  noticeable 
to-day  in  the  practice  of  American  manufacturers,  one  toward  the 
use  of  shorter  teeth  and  the  other  toward  a  larger  angle  of 
obliquity. 

The  effect  of  these  two  changes  upon  the  action  of  gear  teeth 
is  the  subject  of  a  comprehensive  paper  read  by  Mr.  R.  E. 
Flanders  in  1908.1 

Those  readers  who  desire  a  detailed  mathematical  discussion 
of  these  points  are  referred  to  Mr.  Flander's  paper. 

In  brief,  the  effects  of  shorter  teeth  are:  (a)  To  reduce  the 
evils  of  interference  with  involute  teeth;  (b)  to  diminish  the  arc 
of  action;  (c)  to  increase  the  strength;  (d)  to  increase  the  durabil- 
ity; (e)  to  reduce  the  price  of  the  gear. 

The  effects  of  an  increase  in  the  angle  of  obliquity  are  (/) 
to  diminish  interference;  (g)  to  diminish  the  arc  of  action;  (h) 
to  strengthen  the  teeth;  (/)  to  increase  side  pressure  on  bearings; 
(/c)  to  increase  the  lost  work;  (I)  to  distribute  the  wear  more 
evenly. 

It  will  be  noticed  from  the  above  that  the  effects  of  the  two 
changes  are  mainly  the  same.  To  reduce  interference,  to 
strengthen  the  teeth,  and  to  secure  durability  and  uniform  wear 
are  all  desirable  and  important. 

The  question  of  side  pressure  is  not  important  nor  that  of  lost 
work.  The  efficiency  of  accurately  cut  spur  gearing,  according 
to  experiments  by  Lewis  and  others,  is  between  95  and  98  per 
cent. 

Some  examples  of  modern  practice  will  show  the  present 
tendencies.  (See  next  page.) 

Mr.  Flanders  states  that  seven  out  of  eleven  automobile 
manufacturers  questioned  are  using  the  stub  form  of  tooth  and 
like  it. 

1  Trans.  A.  S.  M.  E.,  1908. 


194 


MACHINE  DESIGN 


"\TQTY->O    r\f    •fi-r-m 

Involute 

teeth 

iMame  01  nrm 

Addendum 

Pressure 
angle 

Remarks 

Wm.  Sellers  &  Co  

0  942  m 

20    degrees 

C.  W  Hunt  Co 

0  785  m 

14^  degrees 

Well  man-Sea  ver-M  organ  Co. 
Fellows  Gear  Shaper  Co.  ... 

0.785m 
/  0.70  m 
\0.80m 

20    degrees 
20    degrees 

Steel  mill. 

NOTE. — m  =  module  =  -  • 
n 

A  committee  has  recently  been  appointed  by  the  American 
Society  of  Mechanical  Engineers  to  investigate  the  subject  of 
interchangeable  involute  gearing  and  if  practicable  to  recommend 
a  standard  form.  Mr.  Wilfred  Lewis,  the  chairman  of  the  com- 
mittee, is  on  record  as  approving  a  pressure  angle  of  22J  degrees 
and  an  addendum  of  0.875  m.1 

Mr.  Fellows,  another  member  of  the  committee,  recommends 
an  angle  of  20  degrees  and  a  =  0.75  m. 

Either  of  these  plans  will  do  away  with  interference  and  allow 
of  the  use  of  12-tooth  pinions. 

Mr.  Gabriel  of  the  Brown  &  Sharpe  Manufacturing  Company  is 
in  favor  of  retaining  the  present  angle  of  14^  degrees  and  a  =  m. 

One  objection  to  the  present  standard  is  that  it  is  necessary  to 
empirically  modify  the  addendum  near  the  crest  of  the  tooth  to 
prevent  interference.  It  is  claimed  on  the  other  hand  that 
this  "  easing  off"  of  the  point  of  the  tooth  is  a  help  in  bringing  the 
teeth  together  without  shock. 

Teeth  cut  with  a  milling  cutter  or  planed  by  a  form  can  be 
made  of  empirical  shape  without  difficulty,  but  teeth  generated 
or  "hobbed"  must  correspond  in  all  ways  to  some  theoretical 
curve  which  matches  the  rack  used  as  a  basis  for  the  system. 

At  the  present  writing,  the  problem  of  choosing  an  acceptable 
standard  for  involute  gears  seems  far  from  solution. 


1  Trans.  A.  S.  M.  E.,  1910. 


GEAR  TEETH  195 

96.  Teeth  of  Bevel  Gears. — There  have  been  many  formulas 
and  diagrams  proposed  for  determining  the  strength  of  bevel 
gear  teeth,  some  of  them  being  very  complicated  and  incon- 
venient. It  will  usually  answer  every  purpose  from  a  practical 
standpoint,  if  we  treat  the  section  at  the  middle  of  the  breadth 
of  such  a  tooth  as  a  spur  wheel  tooth  and  design  it  by  the  foregoing 
formulas.  The  breadth  of  the  teeth  of  a  bevel  gear  should  be 
about  one-third  of  the  distance  from  the  base  of  the  cone  to  the 
apex. 

One  point  needs  to  be  noted;  the  teeth  of  bevel  gears  are 
stronger  than  those  of  spur  gears  of  the  same  pitch  and  number  of 
teeth  since  they  are  developed  from  a  pitch  circle  having  an  ele- 
ment of  the  normal  cone  as  a  radius.  To  illustrate,  we  will  sup- 
pose that  we  are  designing  the  teeth  of  a  miter  gear  and  that  the 
number  of  teeth  is  32.  In  such  a  gear  the  element  of  normal  cone 
is  V2  times  the  radius.  The  actual  shape  of  the  teeth  will  then 
correspond  to  those  of  a  spur  gear  having  32V2  =  45  teeth  nearly. 

NOTE. — In  designing  the  teeth  of  gears  where  the  number  is  unknown,  the 
approximate  dimensions  may  first  be  obtained  by  formula  (105)  or  (106)  and 
then  these  values  corrected  by  using  Lewis'  formula. 

PROBLEMS 

1.  The  drum  of  a  hoist  is  8  in.  in  diameter  and  makes  5  revolutions  per 
minute.     The  diameter  of  gear  on  the  drum  is  36  in.  and  of  its  pinion  6  in. 
The  gear  on  the  countershaft  is  24  in.  in  diameter  and  its  pinion  is  6  in.  in 
diameter.     The  gears  are  all  cut. 

Calculate  the  pitch  and  number  of  teeth  of  each  gear,  assuming  a  load  of 
two  tons  on  drum  chain  and  $  =  6000.  Also  determine  the  horse-power  of 
the  machine. 

2.  Calculate  the  pitch  and  number  of  teeth  of  a  cut  cast-steel  gear  10  in. 
in  diameter,  running  at  350  revolutions  per  minute  and  transmitting  20 
horse-power. 

3.  A  cast-iron  gear  wheel  is  30  ft.  6f  in.  in  pitch  diameter  and  has  192 
teeth,  which  are  machine-cut  and  30  in.  broad. 

Determine  the  circular  and  diameter  pitches  of  the  teeth  and  the  horse- 
power which  the  gear  will  transmit  safely  when  making  12  revolutions  per 
minute. 

4.  A  two-pitch  cycloidal  tooth,  6  in.  broad,  72  teeth  to  the  wheel,  failed 
under  a  load  of  38,000  Ib.     Find  value  of  S  by  Lewis'  formula. 

5.  A  vertical  water-whe'el  shaft  is  connected  to  horizontal  head  shaft  by 
cast-iron  gears  and  transmits  150  horse-power.     The  water-wheel  makes 
200  revolutions  per  minute  and  the  head  shaft  100. 


196  MACHINE  DESIGN 

Determine  the  dimensions  of  the  gears  and  teeth  if  the  latter  are  approxi- 
mately two  pitch. 

6.  Work  Problem  1,  uisng  short  teeth  instead  of  standard. 

97.  Rim  and  Arms. — The  rim  of  a  gear,  especially  if  the  teeth 
are  cast,  should  have  nearly  the  same  thickness  as  the  base  of 
tooth,  to  avoid  cooling  strains. 

It  is  difficult  to  calculate  exactly  the  stresses  on  the  arms  of 
the  gear,  since  we  know  so  little  of  the  initial  stress  present,  due 
to  cooling  and  contraction.  A  hub  of  unusual  weight  is  liable 
to  contract  in  cooling  after  the  arms  have  become  rigid  and  cause 
severe  tension  or  even  fracture  at  the  junction  of  arm  and  hub. 

A  heavy  rim  on  the  contrary  may  compress  the  arms  so  as 
actually  to  spring  them  out  of  shape.  Of  course  both  of  thesa 
errors  should  be  avoided,  and  the  pattern  be  so  designed  that 
cooling  shall  be  simultaneous  in  all  parts  of  the  casting. 

The  arms  of  spur  gears  are  usually  made  straight  without 
curves  or  taper,  and  of  a  flat,  elliptical  cross-section,  which  offers 
little  resistance  to  the  air.  To  support  the  wide  rims  of  bevel 
gears  and  to  facilitate  drawing  the  pattern  from  the  sand,  the  arms 
are  sometimes  of  a  rectangular  or  T  section,  having  the  greatest 
depth  in  the  direction  of  the  axis  of  the  gear.  For  pulleys  which 
are  to  run  at  a  high  speed  it  is  important  that  there  should  be  no 
ribs  or  projections  on  arms  or  rim  which  will  offer  resistance  to  the 
air.  Experiments  by  the  writer  have  shown  this  resistance  to  be 
serious  at  speeds  frequently  used  in  practice. 

A  series  of  experiments  conducted  by  the  author  are  reported 
in  the  Am.  Mach.  for  Sept.  22,  1898,  to  which  paper  reference 
is  here  made. 

Twenty-four  pulleys  having  3j  in.  face  and  diameters  of  16, 
20  and  24  in.  were  broken  in  a  testing  machine  by  the  pull  of  a 
steel  belt,  the  ratio  of  the  belt  tensions  being  adjusted  by  levers 
so  as  to  be  two  to  one.  Twelve  of  the  pulleys  were  of  the  ordinary 
cast-iron  type  having  each  six  arms  tapering  and  of  an  elliptic 
section.  The  other  twelve  were  Medart  pulleys  with  steel  rims 
riveted  to  arms  and  having  some  six  and  some  eight  arms.  Test 
pieces  cast  from  the  same  iron  as  the  pulleys  showed  an  average 
modulus  of  rupture  of  35,800  for  the  cast  iron  and  50,800  for  the 
Medart. 

In  every  case  the  arm  or  the  two  arms  nearest  the  side  of  the  belt 


PULLEY  ARMS  197 

having  the  greatest  tension,  broke  first,  showing  that  the  torque 
was  not  evenly  distributed  by  the  rim.  Measurements  of  the 
deflection  of  the  arms  showed  it  to  be  from  two  to  six  times  as 
great  on  this  side  as  on  the  other.  The  buckling  and  springing  of 
the  rim  was  very  noticeable  especially  in  the  Medart  pulleys. 

The  arms  of  all  the  pulleys  broke  at  the  hub  showing  the 
greatest  bending  moment  there,  as  the  strength  of  the  arms  at  the 
hub  was  about  double  that  at  the  rim.  On  the  other  hand,  some 
of  the  cast-iron  arms  broke  simultaneously  at  hub  and  rim, 
showing  a  negative  bending  moment  at  the  rim  about  one-half 
that  at  the  hub. 

The  following  general  conclusions  are  justified  by  these 
experiments: 

(a)  The  bending  moments  on  pulley  arms  are  not  evenly 
distributed  by  the  rim,  but  are  greatest  next  the  tight  side  of 
belt. 

(6)  There  are  bending  moments  at  both  ends  of  arm,  that  at 
the  hub  being  much  the  greater,  the  ratio  depending  on  the 
relative  stiffness  of  rim  and  arms. 

The  following  rules  may  be  adopted  for  designing  the  arms  of 
cast-iron  pulleys  and  gears: 

1.  Multiply  the  net  turning  pressure,  whether  caused  by  belt 
or  tooth,  by  a  suitable  factor  of  safety  and  by  the  length  of  the 
arm  in  inches.     Divide  this  product  by  one-half  the  number  of 
arms  and  use  the  quotient  for  a  bending  moment.     Design  the 
hub  end  of  arm  to  resist  this  moment. 

2.  Make  section  modulus  at  the  rim  ends  of  arms  one-half  as 
strong  as  at  the  hub  ends. 

98.  Sprocket  Wheels  and  Chains. — Steel  chains  connecting 
toothed  wheels  afford  a  convenient  means  of  getting  a  positive 
speed  ratio  when  the  axes  are  some  distance  apart.  There  are 
three  classes  in  common  use,  the  block  chain,  the  roller  chain 
and  the  so-called  " silent"  chain. 

Mr.  A.  Eugene  Michel  publishes  quite  a  complete  discussion 
of  the  design  of  the  first  two  classes  in  Mchy.,  for  February, 
1905,  and  reference  is  here  made  to  that  journal. 

Block  chain  is  that  commonly  used  on  bicycles  and  small 
motor  cars,  so  named  from  the  blocks  with  round  ends  which  are 


198 


MACHINE  DESIGN 


used  to  fill  in  between  the  links.  The  sprocket  teeth  are  spaced 
to  a  pitch  greater  than  that  of  the  chain  links  and  the  blocks 
rest  on  flat  beds  between  the  teeth,  Fig.  89. 

Roller  chains  have  rollers  on  every  pin  and  have  inside  and 
outside  links.  The  sprocket  teeth  have  the  same  pitch  as  the 
chain  Links,  the  rollers  fitting  circular  recesses  between  the 
sprockets,  Fig.  90. 

The  most  serious  failing  of  the  chain  is  its  tendency  to  stretch 
with  use  so  that  the  pitch  becomes  greater  than  that  of  the 
sprocket  teeth. 

To  obviate  this  difficulty  in  a  measure  considerable  clearance 
should  be  given  to  the  sprocket  teeth  as  indicated  in  Fig.  90 
As  the  pitch  of  the  chain  increases  it  will  then  ride  higher  upon 


FIG.  89. 


FIG.  90. 


the  sprockets  until  the  end  of  the  tooth  is  reached.  The  teeth 
are  rounded  on  their  side  faces,  that  they  may  easily  enter  the 
gaps  in  the  chain  and  have  side  clearance. 

Mr.  Michel  gives  the  following  values  for  the  tensile  strength 
of  chains  as  determined  by  actual  tests. 

ROLLER  CHAIN 


Pitch  inches. 

4         f         i 

1 

H 

14 

If 

2  ' 

Tensile 

strength  Ib  . 

1,200 

1,200     4,000 

6,000 

9,000 

12,000 

19,000 

25,000 

BLOCK  CHAIN 


1    inch  pitch  1200  to  2500  Ib. 
1£  inch  pitch  5000  Ib. 


CHAIN  DRIVES  199 

Mr.  Michel  further  recommends  a  factor  of  safety  of  from 
5  to  40  according  to  the  severity  of  the  conditions  as  to  speed  and 
shocks. 

The  tendency  is  to  use  short  links  and  double  or  triple  width 
chains  to  increase  the  rivet  bearing  surface,  as  it  is  this  latter 
factor  which  really  determines  the  life  of  a  chain. 

Roller  chains  may  be  used  up  to  speeds  of  1000  to  1200  ft. 
per  minute. 

The  sprocket  should  be  so  designed  that 'one  tooth  will  carry 
the  load  safely  with  the  pressure  near  the  crest  since  these  con- 
ditions obtain  as  the  chain  stretches.  Use  values  of  S  as  in 
Art.  93. 

99.  Silent  Chains. — The  weak  points  in  the  ordinary  chain, 
whether  it  be  made  with  blocks  or  rollers,  are  the  rivet  bearings. 
It  is  the  continual  wear  of  these,  due  to  insufficient  area  and 
lack  of  proper  lubrication,  that  shortens  the  life  of  a  chain. 

The  so-called  " silent  chain" 
with  rocker  bearings,  is  com- 
paratively free  from  this  defect. 
Fig.  91  illustrates  the  shapes 
of  links,  rivets  and  sprockets  for 
this  kind  of  chain  as  manufac- 
tured by  the  Morse  Chain  Com- 

Pany-  FIG.  91. 

The  chain  proper  is  entirely 

outside  of  the  sprocket  teeth  so  that  the  latter  may  be  contin- 
uous across  the  face  of  the  wheel,  save  for  a  single  guiding 
groove  in  the  center. 

Projections  on  the  under  side  of  the  links  engage  with  the 
teeth  of  the  sprocket,  E  being  the  point  of  contact  for  the  driver 
and  I  a  similar  point  for  the  follower  when  the  rotation  is  as 
indicated. 

Each  rivet  consists  practically  of  two  pins  called  by  the  makers 
the  rocker  pin  and  the  seat  pin.  Each  pin  is  fastened  in  its 
particular  gang  of  links  and  the  relative  motion  is  merely  a 
rocking  of  one  pin  on  the  other  without  appreciable  friction. 

The  pins  are  of  hardened  tool  steel  with  softened  ends.  The 
combination  of  this  freedom  from  rubbing  contact  with  the  adap- 


200 


MACHINE  DESIGN 


tation  of  the  engaging  tooth  profiles,  gives  a  chain  which  can  be 
safely  run  at  high  speeds  without  objectionable  vibration  or 
appreciable  wear. 

The  chains  can  be  made  of  almost  any  width  from  J-  in.  up  to 
18  in.,  the  width  depending  upon  the  pitch  of  the  chain  and  the 
power  to  be  transmitted. 

The  following  are  the  working  loads  -(and  limiting  speeds)  of 
chains  2  in.  in  width  and  of  different  pitches,  taken  from  a  table 
published  by  the  makers: 


Pitch  in  inches  

2 

f 

1 

.9 

1.2 

1.5 

Working  load  in  pounds  

130 

190 

236 

380 

520 

760 

Limiting    speed    revolutions 
per  minute. 

2,000 

1,600 

1,200 

1,100 

800 

600 

The  number  of  teeth  in  the  small  sprocket  may  vary  from  15 
to  30  according  to  the  conditions. 

Assuming  17  teeth  and  the  number  of  revolutions  given  in  the 
above  table  the  speed  of  chain  would  be  1420  ft.  per  minute  for 
the  ^-in.  pitch  and  1275  ft.  per  minute  for  the  1.5  in. 

Chains  of  this  character  have  been  run  successfully  at  2000  ft. 
per  minute. 

PROBLEMS 

1.  Design  eight  arms  of  elliptic  section  for  a  gear  54.  in.  pitch  diameter, 
to  transmit  a  pressure  on  tooth  of  800  Ib.     Material,  cast  iron  having  a  work- 
ing transverse  strength  of  6000  Ib.  per  square  inch. 

2.  Two  sprocket  wheels  of  75  and  17  teeth  respectively  are  to  transmit 
25  horse-power  at  a  chain  speed  of  about  800  ft.  per  minute,  with  a  factor  of 
safety  of  12- 

Determine  the  proper  pitch  of  roller  chain,  the  pitch  diameters  of  the 
sprockets,  and  the  numbers  of  revolutions. 

3.  Suppose  that  in  Problem  2,  a  "  silent "  chain  is  to  be  used  and  the  chain 
speed  increased  to  1200  ft.  per  minute.     Determine  the  proper  pitch  of  chain 
to  be  used  if  the  width  of  chain  is  3  in.     Determine  diameters  and  revolutions 
of  sprockets  as  before. 

100.  Cranks  and  Levers. — A  crank  or  rocker  arm  which  is 
used  to  transmit  a  continuous  or  reciprocating  rotary  motion  is  in 


CRANKS  AND  LEVERS  201 

the  condition  of  a  cantilever  or  bracket  with  a  load  At  the  outer 
end. 

If  the  web  of  the  crank  is  of  uniform  thickness  theory  requires 
that  its  profile  should  be  parabolic  for  uniform  strength,  the 
vertex  of  the  parabola  being  at  the  load  point. 

A  convenient  approximation  to  this  shape  can  be  attained  by 
using  the  tangents  to  the  parabola  at  points  midway  between  the 


d« 


FIG.  92. 

hub  and  the  load  point.  See  Fig.  92.  The  crank  web  is  designed 
of  the  right  thickness  and  breadth  to  resist  the  moment  at  A  B, 
and  the  center  line  is  produced  to  Q,  making  PQ  =  ^PO. 

Straight  lines  drawn  from  Q  to  A  and  B  will  be  tangent  to  the 
parabola  at  the  latter  points  and  will  serve  as  contour  lines  for 
the  web. 

Assume  the  following  dimensions  in  inches: 
1  =  length  of  crank  =  OP 
t  =  thickness  of  web 
h  —  breadth  of  web  =  A  B 
d  =  diameter  of  eye  =  cd 
d±  =  diameter  of  pin 
6  =  breadth  of  eye 
D  =  diameter  of  hub  =  CD 
Dj=  diameter  of  shaft 
B  =  breadth  of  hub. 
If  the  pressure  on  the  crank  pin  is  denoted  by  P  then  will  the 

oment  at  A  B 
section  will  be: 


PI 
moment  at  A  B  be  -     and  the  equations  of  moments  for  the  cross- 


Pl     StW 


2        6  [See  Formula  (3)] 

and  from  this  the  dimensions  at  AB  may  be  calculated. 


202  MACHINE  DESIGN 

The  moment  at  the  hub  will  be  PI  and  will  tend  to  break  the 
iron  on  the  dotted  lines  CD.  The  equation  of  moments  for  the 
hub  is  therefore: 


From  this  equation  the  dimensions  of  the  hub  may  be  calcu- 
lated when  DI  is  known.  The  eye  of  a  crank  is  most  likely  to 
break  when  the  pressure  on  the  pin  is  along  the  line  OP,  and  the 
fracture  will  be  along  the  dotted  lines  cd.  The  bending  moment 
will  be  P  multiplied  by  the  distance  from  center  of  pin  to  center 
of  eye  measured  along  axis  of  pin.  If  we  call  this  distance  x, 
then  will  the  equation  of  moments  be: 

Sb2,, 
Px  =  -^-(d-dl) 

It  is  considered  good  practice  among  engine  builders  to  make 
the  values  of  x,  b  and  B  as  small  as  practicable,  in  order  to  reduce 
the  twisting  moment  on  the  web  of  the  crank  and  the  bending 
moment  on  the  shaft.  In  designing  the  hub,  allowance  must  be 
made  for  the  metal  removed  at  the  key-way. 

PROBLEM 

Design  a  cast-steel  crank  for  a  steam  engine  having  a  cylinder  12  by  30 
in.  and  an  initial  steam  pressure  of  120  Ib.  per  square  inch  of  piston.  The 
shaft  is  6  in.  and  the  crank  pin  3  in.  in  diameter.  The  distance  x  may  be 
assumed  as  4  in.  Calculate, 

1.  Dimensions  of  web  at  AB. 

2.  Dimensions  of  hub  allowing  for  a  key  1  Xf  in. 

3.  Dimensions  of  eye  for  pin  and  make  a  scale  drawing  in  ink  showing 
profile  of  crank  complete.     S  may  be  assumed  as  6000  Ib.  per  square  inch. 

REFERENCES 

Modern  American  Machine  Tools,  Benjamin.     Chapter  VIII. 
Proportions  of  Arms  and  Rims.     Am.  Mack.,  Sept.  30,  1909. 
Efficiency  of  Gears.     Am.  Mach.,  Jan.  12,  1905;  Aug.  19,  1909. 
Strength  of  Gear  Teeth.     Mchy.,  Jan.,  1908;  Am.  Mach.,  Feb.  16,  1905; 

May  9,  1907;  Jan.  16,  1908. 
Proposed  Standard  Systems  of  Gear  Teeth.     Am.  Mach.,  Feb.  25,  1909; 

July  1,  1909. 

Roller  Chains.     Mchy.,  Feb.,  1905. 
Tests  of  Short  Bearings  in  Chains.     Am.  Mach.,  Dec.  28,  1905. 


CHAIN  DRIVES  203 

Progress  in  Chain  Transmission.     Am.  Mach.,  Nov.  7,  1907.  • 
English  Chain  Drives.     Cass.,  May,  1908. 

Lewis'  Experiments  on  Gears.     Tr.  A.  S.  M.  E.,  Vol.  VII,  p.  273. 
Strength  of  Gear  Teeth.     Tr.  A.  S.  M.  E.,  Vol.  XVIII,  p.  766. 
Chain  Gearing.     Tr.  A.  S.  M.  E.,  Vol.  XXIII,  p.  373. 
Symposium  on  Gearing.     Tr.  A.  S.  M.  E.,  Vol.  XXXII,  p.  807. 


CHAPTER  XI 


FLY-WHEELS 

101.  In  General. — The  hub  and  arms  of  a  fly-wheel  are  designed 
in  much  the  same  way  as  those  of  pulleys  and  gears,  the  straight 
arm  with  elliptic  section  being  the  favorite.  The  rims  of  such 
wheels  are  of  two  classes,  the  wide,  thin  rim  used  for  belt  trans- 
mission and  the  narrow  solid  rim  of  the  generator  or  blowing 

engine  wheel.  Fly-wheels  up  to 
8  or  10  ft.  in  diameter  are  usually 
cast  in  one  piece;  those  from  10 
to  16  ft.  in  diameter  may  be  cast 
in  halves,  while  wheels  larger 
than  the  last  mentioned  should 
be  cast  in  sections,  one  arm  to 
FIG.  93.  each  section. 

This  is  a  matter,  not  of  use, 
but  of  convenience  in  casting  and  in  transportation. 

The  joints  between  hub  and  arms  and  between  arms  and  rim 
need  not  be  specially  considered  here,  since  wheels  rarely  fail 
at  these  points. 

The  rim  and  the  joints  -in  the  rim  cannot  be  too  carefully 
designed.  The  smaller  wheel 
cast  in  one  piece  is  more  or 
less  subject  to  stresses  caused 
by  shrinkage.  The  sectional 
wheel  is  generally  free  from 
such  stresses  but  is  weakened 
by  the  numerous  joints. 

Rim  joints  are  of  two  gen- 
eral classes  according  as  bolts 
or  links  are  used  for  fastenings. 

Wide,  thin  rims  are  usually  fastened  together  by  internal 
flanges  and  bolts  as  shown  in  Fig.  93,  while  the  stocky  rims  of 
the  fly-wheels  proper  are  joined  directly  by  links  or  TMiead 
"prisoners"  as  in  Fig.  94. 

204 


FIG.  94. 


FLY  WHEELS  205 

As  will  be  shown  later,  the  former  is  a  weak  antf  unreliable 
joint,  especially  when  located  midway  between  the  arms. 

The  principal  stresses  in  fly-wheel  rims  are  caused  by  centrifu- 
gal force. 

102.  Safe  Speed  for  Wheels.  —  The  centrifugal  force  developed 
in  a  rapidly  revolving  pulley  or  gear  produces  a  certain  tension 
on  the  rim,  and  also  a  bending  of  the  rim  between  the  arms. 
We  will  first  investigate  the  case  of  a  pulley  having  a  rim  of  uni- 
form cross-section. 

It  is  safe  to  assume  that  the  rim  should  be  capable  of  bearing 
its  own  centrifugal  tension  without  assistance  from  the  arms. 

Let     D  =  mean  diameter  of  pulley  rim 
£  =  thickness  of  rim 
b  =  breadth  of  rim 

w  =  weight  of  material  per  cubic  inch 
=  .26  Ib.  for  cast  iron 
=  .28  Ib.  for  wrought  iron  or  steel 
n  =  number  of  arms 
N  =  number  revolutions  per  minute 
t)  =  velocity  of  rim  in  feet  per  second. 

First  let  us  consider  the  centrifugal  tension  alone.  The  cen- 
trifugal pressure  per  square  inch  of  concave  surface  is 

Wv2 


where  W  is  the  weight  of  rim  per  square  inch  of  concave  surface 
=  wt,  and  r  =  radius  in  feet  =  7^7- 

The  centrifugal  tension  produced  in  the  rim  by  this  force  is 
by  formula  (15) 

<j     PD 
S  "IT 

Substituting  the  values  of  p,  W  and  r  and  reducing: 

(110) 


and 


206  MACHINE  DESIGN 

For  an  average  value  of  w  =  .27,  (89)  reduces  to 
S==-JQ  nearly 

a  convenient  form  to  remember. 

The  corresponding  values  of  S  for  dry  wood  and  for  leather 
would  be  nearly: 

Wood  8 

Leather  S  =     . 

If  we  assume  S  as  the  ultimate  tensile  strength,  16,500  Ib. 
for  cast  iron  in  large  castings  and  60,000  Ib.  for  soft  steel,  then 
the  bursting  speed  of  rim  is: 

for  a  cast-iron  wheel  r  =  406  ft.  per  second  (112) 

and  for  steel  rim  v  =  775  ft.  per  second  (113) 

and  these  values  may  be  used  in  roughly  calculating  the  safe 
speed  of  pulleys. 

It  has  been  shown  by  Mr.  James  B.  Stanwood,  in  a  paper  read 
before  the  American  Society  of  Mechanical  Engineers,1  that  each 
section  of  the  rim  between  the  arms  is  moreover  in  the  condition 
of  a  beam  fixed  at  the  ends  and  uniformly  loaded. 

This  condition  will  produce  an  additional  tension  on  the  outside 
of  rim.  The  formula  for  such  a  beam  when  of  rectangular  cross- 
section  is 

Wl     Sbd2 


12        6 


(b) 


W  in  this  case  is  the  centrifugal  force  of  the  fraction  of  rim 
included  between  two  arms. 

The  weight  of  this  fraction  is  -       -  and  its  centrifugal  force 


n          gD  gn 

Also  /  —  —  and  d  =  t 

n 

'  See  Trans.  A.  S.  M.  E.,  Vol.  XIV. 


FLY  WHEELS  207 

Substituting  these  values  in  (b)  and  solving  for  $? 

£  =  3.678—^  (c) 

If  w  is  given  an  average  value  of  .27  then 

S  =  T^nearly  (d) 


and  the  total  value  of  the  tensile  stress  on  outer  surface  of  rim  is 
Solving  for  v: 


9) 

nearly.  (114) 


V 


& 


£>_L'  <115) 

tn2     10 

In  a  pulley  with  a  thin  rim  and  small  number  of  arms,  the 
stress  due  to  this  bending  is  seen  to  be  considerable. 

It  must,  however,  be  remembered  that  the  stretching  of  the 
arms  due  to  their  own  centrifugal  force  and  that  of  the  rim  will 
diminish  this  bending.  Mr.  Stanwood  recommends  a  deduction 
of  one-half  from  the  value  of  S  in  (d)  on  this  account. 

Prof.  Gaetano  Lanza  has  published  quite  an  elaborate  mathe- 
matical discussion  of  this  subject.  (See  Vol.  XVI,  Trans. 
A.  S.  M.  E.)  He  shows  that  in  ordinary  cases  the  stretch  of  the 
arms  will  relieve  more  than  one-half  of  the  stress  due  to  bending, 
perhaps  three-quarters. 

103.  Experiments  on  Fly-wheels. — In  order  to  determine 
experimentally  the  centrifugal  tension  and  bending  in  rapidly 
revolving  rims,  a  large  number  of  small  fly-wheels  have  been 
tested  to  destruction  at  the  Case  School  laboratories.  In  all 
ten  wheels,  15  in.  in  diameter  and  twenty-three  wheels  2  ft.  in 
diameter  have  been  so  tested.  An  account  of  some  of  these 
experiments  may  be  found  in  Trans.  A.  S.  M.  E.,  Vol.  XX. 
The  wheels  were  all  of  cast  iron  and  modeled  after  actual  fly- 
wheels. Some  had  solid  rims,  some  jointed  rims  and  some  steel 
spokes. 

To  give  to  the  wheels  the  speed  necessary  for  destruction, 
use  was  made  of  a  Dow  steam  turbine  capable  of  being  run  at  any 
speed  up  to  10,000  revolutions  per  minute.  The  turbine  shaft 


208 


MACHINE  DESIGN 


was  connected  to  the  shaft  carrying  the  fly-wheels  by  a  brass 
sleeve  coupling  loosely  pinned  to  the  shafts  at  each  end  in  such  a 
way  as  to  form  a  universal  joint,  and  so  proportioned  as  to  break 
or  slip  without  injuring  the  turbine  in  case  of  sudden  stoppage 
of  the  fly-wheel  shaft. 

One  experiment  with  a  shield  made  of  2-in.  plank  proved  that 
safety  did  not  lie  in  that  direction,  and  in  succeeding  experiments 
with  the  15-in.  wheels  a  bomb-proof  constructed  of  6Xl2-in. 
white  oak  was  used.  The  first  experiment  with  a  24-in.  wheel 
showed  even  this  to  be  a  flimsy  contrivance.  In  subsequent 
experiments  a  shield  made  of  12Xl2-in.  oak  was  used.  This 
shield  was  split  repeatedly  and  had  to  be  re-enforced  by  bolts. 

A  cast-steel  ring  about  4  in.  thick,  lined  with  wooden  blocks 
and  covered  with  3-in.  oak  planking,  was  finally  adopted. 

The  wheels  were  usually  demolished  by  the  explosion.  No 
crashing  or  rending  noise  was  heard,  only  one  quick,  sharp  report, 
like  a  musket  shot. 

The  following  tables  give  a  summary  of  a  number  of  the 
experiments. 

TABLE  L 

FIFTEEN-INCH  WHEELS 


Bursting  speed 

Centrifugal 

No. 

tension 

<ii2 

Remarks 

Rev. 

Feet  per 

V 

in 

per  minute           second  =  v 

1               6,525 

430                     18,500 

Six  arms. 

2               6,525 

430                     18,500 

Six  arms. 

3               6,035 

395                     15,600 

Thin  rim. 

4               5,872                        380                     14,400 

Thin  rim. 

5              2,925                        192                      3,700 

Joint  in  rim. 

6               5,600  l                     368                     13,600 

Three  arms. 

7 

6,198 

406                     16,500 

Three  arms. 

8 

5,709 

368 

13,600 

Three  arms. 

9 

5,709 

365 

13,300 

Thin  rim. 

10 

5,709                        361 

13,000 

Thin  rim. 

1  Doubtful. 


FLY  WHEELS 


209 


TABLE  LI 

TWENTY-FOUR-INCH    WHEELS 


No. 

Shape  and  size  of  rim 

Weight 
of 
wheel, 
pounds 

Diam- 
eter, 
inches 

Breadth 
inches 

Depth, 
inches 

Area, 
square 
inches 

Style  of  joint 

11 
12 
13 
14 
15 
16 
17 

24 
24 
24 
24 
24 
24 
24 

2£ 

4  A 

4 

4 

4  A 

1.2 
1.2 

1.5 
.75 
.75 
.75 
.75 
2.1 
2.1 

3.18 
3.85 
3.85 
3.85 
3.85 
2.45 
2.45 

Solid  rim  

75.25 
93. 
91.75 
95. 
94.75 
65.1 
65. 

Internal  flanges,  bolted  
Internal  flanges,  bolted  
Internal  flanges,  bolted  
Internal  flanges,  bolted  
Three  lugs  and  links.  . 

Two  lugs  and  links  

TABLE  LII 
FLANGES  AND  BOLTS 


Flanges 

Bolts 

No. 

Thickness, 
inches 

Effective 
breadth, 
inches 

Effective 
area, 
inches 

No.  to  each 
joint 

Diameter, 
inches 

Total  tensile 
strength, 
pounds 

12 

u 

2.8 

1.92 

4 

Ire- 

16,000 

13 

li 

2.75 

1.89 

4 

& 

16,000 

14 

H 

2.75 

2.58 

4 

A 

16,000 

15 

il 

2.5 

2.34 

4 

I 

20,000 

BY    TESTING    MACHINE 


Tensile  strength  of  cast  iron 
Transverse  strength  of  cast  iron 
Tensile  strength  of  y^r  bolts 
Tensile  strength  of  £  bolts 


=  19,600  Ib.  per  square  inch. 
46,600  Ib.  per  square  inch. 
=   4,000  Ib. 
5,000  Ib. 


210 


MACHINE  DESIGN 


TABLE  LIII 
FAILURE  OF  FLANGED  .JOINTS 


Area  of 

Effect 
area 

Total 

Bursting 
speed 

Cent,  tension 

No. 

rim, 

flanges, 

strength 

Remarks 

square 
inches 

square 
inches 

bolts, 
pounds 

Rev. 

per 

Ft.  per 
sec. 

Per 

sq.  in. 

Total 

min. 

=  v 

=  10 

11 

3.18 

3  672 

385 

14  800 

47  000 

12 

3.85 

1.92 

16  000 

13 

3.85 

1.89 

16,000 

1,760 

184 

3,400 

13,100 

Flange  broke. 

14 

3.85 

2.58 

16,000 

1,875 

196 

3,850 

14,800 

Bolts  broke. 

15 

3.85 

2.34 

20,000 

1,810 

190 

3,610 

13,900 

Flange  broke. 

TABLE  LIV 
LINKED  JOINTS 


No. 

Lugs 

Links 

Rim 

Breadth 
inches 

Length 
inches 

Area, 
sq.  in. 

Number 
used 

Effect 
breadth, 
inches 

Thick- 
ness, 
inches 

Effective 
area, 
sq.  in. 

Max. 
area, 
sq.  in. 

Net 
area, 
sq.  in. 

16 
17 

.45 

.44 

1.0 

.98 

.45 
.43 

3 
2 

.57 
.54 

.327 
.380 

.186 
.205 

2.45 
2.45 

1.98 
1.98 

BY    TESTING    MACHINE 

Tensile  strength  of  cast  iron  =  19,600. 
Transverse  strength  of  cast  iron  =40,400. 
Av.  tensile  strength  of  each  link  =  10, 180. 

TABLE  LV 

FAILURE  OF  LINKED  JOINTS 


Bursting  speed 

Cent,  tension 

Strength 

Strength 

No. 

of  links, 

of  rim, 

Remarks 

pounds 

pounds 

Rev.  per 

Ft.  per 

Per  sq.  in. 

V2 

Total 

mm. 

sec.  =  v 

10 

16 
17 

30,540 
20,360 

38,800 
38,800 

3,060 
2,750 

320 
290 

10,240 
8,410 

25,100 
20,600 

Rim  broke. 
Lugs     and     rim 
broke. 

FLY  WHEELS  211 

The  flanged  joints  mentioned  had  the  internal  flanges  and  bolts 
common  in  large  belt  wheel  rims  while  the  linked  joints  were  such 
as  are  common  in  fly-wheels  not  used  for  belts. 

Subsequent  experiments1  have  given  approximately  the  same 
results  as  those  ju&t  detailed.  The  highest  velocity  yet  attained 
has  been  424  ft.  per  second;  this  is  in  a  solid  cast-iron  rim  with 
numerous  steel  spokes.  The  average  bursting  velocity  for  solid 
cast  rims  with  cast  spokes  is  400  ft.  per  second. 

Wheels  with  jointed  rims  burst  at  speeds  varying  from  190  to 
250  ft.  per  second,  according  to  the  style  of  joint  and  its  location. 
The  following  general  conclusions  seem  justified  by  these  tests. 

1 .  Fly-wheels  with  solid  rims,  of  the  proportions  usual  among 
engine  builders  and  having  the  usual  number  of  arms,  have  a 
sufficient  factor  of  safety  at  a  rim  speed  of  100  ft.  per  second  if 
the  iron  is  of  good  quality  and  there  are  no  serious  cooling  strains. 

In  such  wheels  the  bending  due  to  centrifugal  force  is  slight, 
and  may  safely  be  disregarded. 

2.  Rim  joints  midway  between  the  arms  are  a  serious  defect  and 
reduce  the  factor  of  safety  very  materially.     Such  joints  are  as 
serious  mistakes  in  design  as  would  be  a  joint  in  the  middle  of  a 
girder  under  a  heavy  load. 

3.  Joints  made  in  the  ordinary  manner,  with  internal  flanges 
and  bolts,  are  probably  the  worst  that  could  be  devised  for  this 
purpose.     Under  the  most  favorable  circumstances  they  have 
only  about  one-fourth  the  strength  of  the  solid  rim  and  are  par- 
ticularly weak  to  resist  bending. 

See  Fig.  95,  which  shows  the  opening  of  such  a  joint  and  the 
bending  of  the  bolts. 

In  several  joints  of  this  character,  on  large  fly-wheels,  calcu- 
lation has  shown  a  strength  less  than  one-fifth  that  of  the  rim. 

4.  The  type  of  joint  known  as  the  link  or  prisoner  joint  is 
probably  the  best  that  could  be  devised  for  narrow  rimmed 
wheels  not  intended  to  carry  belts,  and  possesses,  when  properly 
designed,   a  strength  about  two-thirds  that  of  the  solid  rim. 

In  1902-04  experiments  on  four-foot  pulleys  were  conducted 
by  the  writer,  and  the  results  published.2 

A  cast-iron,  whole  rim  pulley  48  in.  in  diameter,  burst  at  1100 

1  Trans.  A.  S.  M.  E.,  Vol.  XXIII. 

2  Trans.  A.  S.  M.  E.,  Vol.  XXVI. 


212  MACHINE  DESIGN 

revolutions  per  minute  or  a  linear  speed  of  230  ft.  per  second, 
the  rupture  being  caused  by  a  balance  weight  of  3^  Ib.  which  had 
been  riveted  inside  the  rim  by  the  makers.  The  centrifugal 
force  of  this  weight  at  1100  revolutions  per  minute  was  2760  Ib. 

A  cast-iron  split  pulley  of  the  same  dimensions  burst  at  a  speed 
of  about  600  revolutions  per  minute,  or  a  linear  speed  of  only  125 
ft.  per  second. 

The  failure  was  due  to  the  unbalanced  weight  of  the  joint 
flanges  and  bolts  which  were  located  midway  between  the  arms. 
Such  a  pulley  is  not  safe  at  high  belt  speeds. 

104.  Wooden  Pulleys.  —  Experiments  on  the  bursting  strength 
of  wooden  pulleys  were  conducted  at  the  Case  School  laboratories 
in  1902-3  under  the  writer's  direction.1 

These  are  of  some  interest  in  view  of  the  use  of  this  material 
for  fly-wheel  rims.  As  noted  in  Art.  102,  the  tensile  stress  in 
wood  due  to  the  centrifugal  force  is  only  ^  that  of  cast  iron  under 
similar  circumstances.  Assuming  the  tensile  strength  of  the 
wood  to  be.  10,000  Ib.  per  square  inch,  and  substituting  this  value 

in  the  equation  S  =       .  we  have  the  bursting  speed  of  a  wooden 


pulley  #  =  1000  ft.  per  second  nearly. 

This  for  wood  without  joints. 

The  24-in.  pulleys  tested  had  wood  rims  glued  up  in  the  usual 
manner  and  jointed  at  two  opposite  points.  The  wheels  burst 
at  speeds  varying  from  1700  to  2450  revolutions  per  minute,  or 
linear  rim  speeds  varying  from  178  to  257  ft.  per  second,  thus 
comparing  favorably  with  cast-iron  split  pulleys.  The  rims 
usually  failed  at  the  points  where  the  arms  were  mortised  in,  and 
the  stiffening  braces  at  these  points  did  more  harm  than  good. 
A  wooden  pulley  with  solid  rim  and  web  remained  intact  at 
4450  revolutions  per  minute,  or  467  ft.  per  second,  a  higher  speed 
than  that  of  any  cast-iron  pulley  tried. 

105.  Rims  of  Cast-iron  Gears.  —  A  toothed  wheel  will  burst  at 
a  less  speed  than  a  pulley  because  the  teeth  increase  the  weight 
and  therefore  the  centrifugal  force  without  adding  to  the  strength. 

The  centrifugal  force  and  therefore  the  stresses  due  to  the  force 

1  Mchy.,  N.  Y.,  Aug.,  1905. 


FLY  WHEELS 


213 


214 


MACHINE  DESIGN 


will  be  increased  nearly  in  the  ratio  that  the  weight  of  rim  and 
teeth  is  greater  than  the  weight  of  rim  alone. 

This  ratio  in  ordinary  gearing  varies  from  1.5  to  1.7.  We  will 
assume  1.6  as  an  average  value.  Neglecting  bending  we  now 
have  from  equation  (110) 


(116) 


and 


=  326.2  ft.  per  second 


Including  bending 


S'  =  i.6r*  (--+,- 
?     10 


(117) 
(118) 


As  the  transverse  strength  of  cast  iron  by  experiment  is  about 
double  the  tensile  strength,  a  larger  value  of  S  may  be  allowed  in 
formulas  (114)  (115)  (118). 

In  built-up  wheels  it  is  better  to  have  the  joints  come  near 
the  arms  to  prevent  the  tendency  of  the  bending  to  open  the 

joints,  and  the  fastenings 
should  have  the  same  tensile 
strength  as  the  rim  of  the 
wheel. 

106.  Rotating  Discs.— The 
formulas  derived  in  Art.  102 
will  only  apply  in  the  case  of 
thin  rims  and  cannot  be  used 
for  discs  or  for  rims  having 
any  considerable  depth.  The 
determination  of  the  stresses 
in  a  rotating  disc  is  a  com- 
plicated and  difficult  problem,  if  the  material  is  regarded  as 
perfectly  elastic. 

A  rational  solution  of  this  problem  may  be  found  in  Stodola's 
Steam  Turbines,  pp.  157-69.  For  the  purposes  of  this  treatise 
an  approximate  solution  is  preferred,  the  elasticity  of  the  metal 
being  neglected.  This  method  of  treatment  is  much  simpler, 


FIG.  96. 


ROTATING  DISCS  215 

and  as  the  metals  used  are  imperfectly  elastic  (especially  the 
cast  metals)  the  results  obtained  will  probably  be  as  reliable  as 
any  —  for  practical  use. 

The  following  discussion  is  an  abstract  of  one  given  by  Mr. 
A.  M.  Levin  in  the  Am.  Mach.1  the  notation  being  changed 
somewhat. 

107.  Plain  Discs.  —  Let  Fig.  96  represent  a  ring  of  uniform 
thickness  t,  having  an  external  diameter  D  and  an  internal 
diameter  d,  all  in  inches. 

Let  v  =  external  velocity  in  feet  per  second 

Let  a  =  angular  velocity  =-jr- 

r  =  radius  to  center  of  gravity  of  half  ring  in  feet 
w  =  weight  of  metal  per  cubic  inch. 
The  value  of  r  for  a  half-ring  is  easily  proved  to  be  : 

2      DB-d5  . 


or  J_     D*-d*  . 

~'  U 


The  weight  of  the  half-ring  is: 
W=^(D 

o 

and  its  centrifugal  force  : 

W<?r_ 

~7~          ~144T 
Substituting  for  a  its  value  in  terms  of  v: 


Now  if  we  assume  the  stress  on  the  area  at  AB  due  to  the 
centrifugal  force  to  be  uniformly  distributed:  (and  here  lies  the 
approximation)  then  will  the  tensile  stress  on  the  section  be 


C      _ 

~(D-d)t~  * 


Am.  Mach.,  Oct.  20,  1904. 


216 


MACHINE  DESIGN 


For  a  solid  disc: 


For  a  thin  ring: 


9 

12wv2 


(122) 


(123) 


or  the  same  as  in  equation  (110). 

If  the   metal   be  perfectly   elastic,   Stodola's  formulas  give 

riiyi) 
S  = as  the  stress  near  the  center  when  d  approaches  0  — 

or  more  than  twice  the  value  given  in  (122).  In  view  of  the 
imperfect  elasticity  of  the  metals  used  the  true  value  will  prob- 
ably be  between  these  two.  This  value  should  be  determined 
by  experiment. 

108.  Conical  Discs. — Let  Fig.  97  represent  a  ring  whose  thick- 
ness varies  uniformly  from  the  inner  to  the  outer  circumference 
and  whose  dimensions  are  as  follows: 


/ 

Al 

|l 
|| 

D 

V 

\ 
\ 

II 
|l 

/ 

1 

FIG.  97. 

D  =  outer  diameter  in  inches 

d  =  inner  diameter  in  inches 

6  =  breadth  of  ring  at  inner  circumference 
m  =  tangent  of  angle  of  slant  CAD 

D-d  D-d 

Then  m  =  — =- —  or  b  =  —  — 
o  m 


ROTATING  DISCS  217 

By  cutting  the  ring  into  slices  perpendicular  to  the  axis, 
finding  the  centrifugal  force  for  each  slice  and  then  integrating 
between  D  and  d,  the  centrifugal  force  of  the  half-ring  is  found 
to  be: 


C  =  -  ^-  (124) 

mgD2 

The  area  on  the  line  A B  to  resist  the  centrifugal  force  is: 

(125) 


or  a  stress  one-half  that  of  a  plain  flat  disc. 

109.  Discs  with  Logarithmic  Profile.  —  A  form  of  disc  some- 
times used  for  steam  turbines  consists  of  a  solid  of  revolution 
generated  by  a  curve  of  the  equation 


revolving  around  the  x—  axis. 

Mr.  Levin  investigates  two  curves  of  this  character: 

y  =  \og  x  and  y=2  log  | 
and  finds  the  stresses  to  be  respectively: 
Whena-6  5  =  1.5—-  (127) 


When  a-  §6  S  =  1.2—  •  (128) 

<J 

The  general  equation  for  S  in  this  case  is: 

«-96^.£  (129) 

and  in  deriving  the  formulas   (127)   and   (128)   D  is  assumed 
as  8a  and  as  9a  respectively. 


218 


MACHINE  DESIGN 


110.  Bursting  Speeds. — It  will  be  seen  that  all  the  formulas 
for  centrifugal  stress  may  be  reduced  to  the  general  form: 


S  =  k 


wv* 
9 


(130) 


where  &  is  a  constant  depending  upon  the  shape  of  the  rotating 
body. 


The  following  table  gives  the  values  of  v  =  A  y -,    the    burst- 

\kw 

ing    speed   of  rim  in  feet  per  second,  for   different    materials 
and  different  shapes. 

TABLE  LVI 

BUKSTING  SPEEDS  IN  FEET  PER  SECOND 


Weight 

Values  of  v 

Metal 

per 
cubic 
inch 

strength 

Thin 
ring 

Perforated 
disc 
(Stodola) 

Flat 
disc 

Taper 
disc 

Logar- 
ithmic 
disc 

w 

S 

ft-  12 

fc  =  9 

&  =  4 

k  =  2 

&  =  1.5 

Cast  iron  

.26 

18,000 

430 

500 

745 

1,050 

1,215 

Manganese  bronze  .... 

.315 

60,000 

715 

825 

1,240 

1,750 

2,050 

Soft  steel  

.28 

60,000 

760 

880 

1,315 

1,860 

2,140 

111.  Tests  of  Discs. — During  the  years  1906-07,  certain  tests 
were  made  on  cast-iron  discs  in  the  laboratories  of  the  Case 
School  of  Applied  Science  by  senior  students,  Messrs.  Baxter, 
Brown,  Goss  and  Jeffrey. 

The  discs  experimented  upon  were  from  16  to  18  in.  in  diameter 
and  from  J  to  1  in.  in  thickness  and  were  cast  from  a  soft  gray 
iron,  clean  and  free  from  defects.  '  The  average  tensile  strength 
of  the  iron  was  15,750  Ib.  per  square  inch  and  the  transverse 
strength  37,800  Ib.  per  square  inch.  All  of  the  discs  were  finished 
to  insure  good  balancing. 

They  were  tested  to  bursting  by  centrifugal  force  with  the 
apparatus  before  described  in  the  article  on  Fly-wheels,  the 
speed  being  measured  by  a  reducing  gear  and  counter.  Each  of 
the  discs  had  a  1-in.  hole  through  the  center;  some  had  hubs  2  in. 


ROTATING  DISCS 


219 


in  diameter  and  some  were  plain  as  noted  in  the  following  table 
which  gives  a  resume  of  the  results. 

TABLE  LVII 

BURSTING  SPEED  OF  CAST-IRON  Discs 


Calculated 

Diameter, 
inches 

Weight, 
pounds 

Thickness, 
inches 

Length  of 
hub, 
inches 

Bursting 
speed 
r.p.m. 

Velocity  of 
rim,  ft.  per  sec. 

k-S^ 
'     wv* 

18 

.418 

2.00 

7,755 

610 

5.25 

18 

.775 

2.00 

7,125 

560 

6.24 

18 

.573 

2.00 

8,700 

683 

4.19 

16 

28.75 

.562 

2.125 

9,282 

650 

4.62 

16 

27.25 

.514 

2.125 

9,486 

660 

4.49 

16 

55.50 

1.086 

None 

9,690 

676 

4.28 

16 

48.00 

.951 

None 

8,262 

577 

5.86 

18 

61.25 

.953 

None 

8,364 

656 

4.55 

18 

47.75 

.715 

2.00 

9,180 

720 

3.76 

16 

42.00 

.820 

1.57 

8,874 

620 

5.08 

16                   45.00 

.873               1.62 

9,792 

685 

4.16 

18                    65.50      :         .961 

1.85 

9,792 

770                   3.29 

Average  value  of  k  =  4.64. 

The  presence  or  absence  of  a  hub  has  no  apparent  effect  on  the  strength.     The  value  of 
k,  as  was  to  have  been  expected  is  slightly  greater  than  the  4  given  in  formula  (122). 

PROBLEMS 

1.  Determine  bursting  speed  in  revolutions  per  minute,  of  a  gear  42  in. 
in  diameter  with  six  arms,  if  the  thickness  of  rim  is  .  75  in. 

(1)  Considering  centrifugal  tension  alone. 

(2)  Including  bending  of  rim  due  to  centrifugal  force  assuming  that 
three-fourths  the  stress  due  to  bending  is  relieved  by  the  stretching  of  the 
arms. 

2.  Design  a  link  joint  for  the  rim  of  a  fly-wheel,  the  rim  being  8  in.  wide, 
12  in.  deep  and  18  ft.  mean  diameter,  the  links  to  have  a  tensile  strength  of 
65,000  Ib.  per  square  inch.     Determine  the  relative  strength  of  joint  and  the 
probable  bursting  speed. 

3.  Discuss  the  proportions  of  one  of  the  following  wheels  in  the  laboratory 
and  criticise  dimensions. 

(a)  Fly-wheel,  Allis  engine. 

(b)  Fly-wheel,  Fairbanks  gas  engine. 

(c)  Fly-wheel,  air  compressor. 

(d)  Fly-wheel,  Buckeye  engine. 

(e)  Fly-wheel,  pumping  engine. 

4.  Determine  the  value  of  C  in  formula  (124)  by  calculation. 

5.  A  Delaval  turbine  disc  is  made  of  soft  steel  in  the  shape  of  the  logarith- 


220  MACHINE  DESIGN 

mic  curve  without  any  hole  at  the  center.     Determine  the  probable  bursting 
speed  if  the  disc  is  12  in.  in  diameter. 

6.  A  wheel  rim  is  made  of  cast  iron  in  the  shape  of  a  ring  having  diameters 
of  4£  ft.  and  6  ft.,  inside  and  outside.  Determine  probable  bursting  speed. 

7.  Substitute  the  value  for  centrifugal  force  in  place  of  internal  pressure 
in  Barlow's  formula  (b)  Art.  22,  and  derive  a  value  for  S  in  a  rotating  ring. 

Test  this  f or  d  =  -  and  compare  with  formulas  in  preceding  article. 

REFERENCES 

Resistance  of  Air  to  Fly-wheels.     Cass.,  Feb.,  1903;  Mar.,  1903. 

Shields  to  Reduce  Windage.     Power,  Dec.,  1903. 

Testing  Pit  at  Purdue  University.     Am.  Mach.,  Aug.  26,  1909. 

High  Efficiency  Joint  for  Rim.     Am.  Mach.,  Feb.  28,  1907;  Apr.  4,  1907; 

Apr.  11,  1907. 
Rolling  Mill  Fly-wheels.     Tr.  A.  S.  M.  E.,  Vol.  XX,  p.  944. 


CHAPTER  XII 
TRANSMISSION  BYl  BELTS  AND*  ROPES 

112.  Friction  of  Belting. — The  transmitting  power  of  a  belt  is 
due  to  its  friction  on  the  pulley,  and  this  friction  is  equal  to  the 
difference  between  the  tensions  of  the  driving  and  slack  sides  for 
the  belt.  <j- 

Let  w  =  width  of  belt  /\d 


7\—  tension  of  driving  side 
T2  =  tension  of  slack  side 
R=  friction  of  belt 


>0 


=  T!-T,  ^     \ 

/=  coefficient  of  friction  between  *^"V    / 

belt  and  pulley  \fB 

6  =  arc    of    contact    in    circular  ^~ 

FIG.  98. 
measure. 

The  tension  T  at  any  part  of  the  arc  of  contact  is  intermediate 
between  Tl  and  T2. 

Let  AB  Fig.  98  be  an  indefinitely  short  element  of  the  arc  of 
contact,  so  that  the  tensions  at  A  and  B  differ  only  by  the 
amount  dT. 

dT  will  then  equal  the  friction  on  A  B  which  we  may  call  dR. 

Draw  the  intersecting  tangents  OT  and  OT'  to  represent  the 
tensions  and  find  their  radial  resultant  OP.  Then  will  OP 
represent  the  normal  pressure  on  the  arc  A  B  which  we  will  call  P. 

<OTP  =  <ACB  =  dO 

.'.  P  =  TdO. 
The  friction  on  A  B  is 

fP  =fTd0 

dT 

and  Jd0  =  -^j- 

1 

221 


222  MACHINE  DESIGN 

Integrating  for  the  whole  arc  6: 

*dT  T, 


R  =  T,-  T2  =  7\(1-  e~f°).  (131) 

The  value  of  /  varies  with  the  nature  of  the  materials  used,  the 
tension  and  slip  of  the  belt  and  the  speed  of  the  pulleys.  If  we 
denote  the  expression  (1  —e—fd)  by  C,  then  for  different  values 
of  /  and  the  arc  of  contact,  C  has  the  following  values. 

ARC  OF  CONTACT 


Values  of/ 

90 

120 

150 

180 

200 

.20 

.270 

.342 

.408 

.467 

.503 

.25 

.325 

.407 

.480 

.544 

.582 

.30 

.376 

.467 

.544 

.610 

.649 

.35 

.423 

.520 

.600 

.667 

.705 

.40 

.467 

.567 

.649 

.715 

.753 

.45 

.507 

.610 

.692 

.757 

.792 

The  friction  or  force  transmitted  by  a  belt  per  inch  of  width 
is  then 

R  =  CTl  (132) 

and  Tl  must  not  exceed  the  safe  working  tensile  strength  of  the 
material. 

A  handy  rule  for  calculating  belts  assumes  C  — .5  which  means 
that  the  force  which  a  belt  will  transmit  under  ordinary  condi- 
tions is  one-half  its  tensile  strength. 

The  conditions  assumed  above  are  only  average  ones  and  the 
formulas  are  only  approximate  for  any  particular  case.  The 
coefficient  of  friction  varies  with  the  materials  used  for  pulleys 
and  belts,  with  the  tension,  the  speed  and  the  amount  of  slip. 


BELTS  223 

The  sum  of  the  tensions  is  not  constant  as  may  be  readily  proved 
by  experiment.  Mr.  Barth  shows  from  theoretical  considerations 
of  the  elasticity  of  the  belt  that  approximately:1 


where  T0  =  initial  tension  (unloaded),  or  that  the  sum  of  the 
square  roots  of  the  two  tensions  is  a  constant.  For  instance,  if 
T0  =  100,  we  have 


If  we  assume  values  for  Tl  and  solve  for  T2,  we  have: 


7\  T2  Tl 

121  81  202 

144  64  208 

169  49  218 

196  36  232 

and  the  sum  of  the  tensions  increases  as  the  load  increases. 

113.  Slip  of  Belt.  —  Mr.  Wilfred  Lewis  in  his  experiments  on 
belts2  found  that  the  coefficient  of  friction  varied  with  the  slip, 
increasing  as  the  slip  increased,  so  that  as  the  load  became 
heavier  the  slipping  of  the  belt  increased  its  driving  power  and 
prevented  further  slip. 

A  distinction  must  be  made  between  slip  due  to  the  load  and 
slip,  or  "  creep  "  as  it  is  usually  called,  due  to  the  stretching  of  the 
belt. 

As  has  been  already  explained,  the  tension  of  a  belt  varies  in 
passing  over  the  driving  pulley  from  T1  to  T2  and  in  passing  over 
the  driven  pulley  from  T2  to  Tr  The  belt  is  elastic  and  stretches 
more  or  less  according  to  the  tension,  so  that  its  length  is  con- 
tinually changing  as  it  passes  over  either  pulley.  This  produces 
a  "  creep  "  or  relative  motion  of  the  belt  on  the  pulley,  positive 
on  one  pulley  and  negative  on  the  other;  i.e.,  the  belt  gains  on  the 
driven  pulley  and  loses  on  the  driving  pulley. 

Experiments  by  Professor  Bird3  show  a  creep  under  ordinary 

1  Trans.  A.  S.  M.  E.,  1909. 

2  Trans.  A.  S.  M.  E.,  1886. 

3  Trans.  A.  S.  M.  E.,  1905. 


224  MACHINE  DESIGN 

conditions  of  about  1  per  cent  and  a  working  modulus  of  elasticity 
for  leather  belting  of  from  12;000  to  30,000  with  an  average  of 
20,000.  Slip  due  to  increase  of  load  will  be  added  to  the  creep. 

Tests  of  belting  reported  in  1911  by  Professor  W.  M.  Sawdon1 
indicate  a  marked  variation  in  the  slip  of  belts  without  any 
apparent  change  in  the  conditions. 

With  the  belt  tension,  the  load  and  the  speed  remaining  the 
same,  the  slip  would  sometimes  remain  constant  at  1  or  2  per 
cent  for  30  or  35  minutes  and  then  suddenly  rise  to  10  or  15  per 
cent. 

In  these  experiments  it  was  found  that  the  load  capacity  of  a 
leather  belt  on  pulleys  of  various  materials  was  as  follows,  cast 
iron  being  taken  as  a  standard: 

Cast  iron  ......................................    100 

Wood  .........................................   105 

Paper  .........................................    137 

The  effect  of  cork  inserts  was  to  increase  the  driving  capacity 
of  the  cast-iron  pulleys  10  to  12  per  cent.  The  wood  pulleys 
received  no  benefit  from  cork  inserts  while  the  capacity  of  the 
paper  pulleys  was  diminished  by  the  cork. 

The  wood  pulleys  showed  a  small  overload  capacity,  being 
inferior  to  the  cast  iron  at'slips  exceeding  3  or  4  per  cent. 

114.  Coefficient  of  Friction.  —  Mr.  Barth,  as  a  result  of  the 
experiments  of  Mr.  Lewis  and  an  exhaustive  study  of  the  whole 
subject,  suggests  the  following  formulas  for  the  coefficient: 

(133) 


where  v  is  the  average  velocity  of  sliding  of  belt  on  pulley  or 
one-half  the  total  slip  in  feet  per  minute  and  V  is  the  velocity 
of  belt  in  feet  per  minute. 

1  Proc.  Nat.  Ass'n  of  Cotton  Manufacturers,  Sept.,  1911. 


SLIP  OF  BELTS  225 

Values  of  /  from  equation  (133)   are: 

/= 

2  0.267  See  Art.    112. 

4  0.350 

6  0.400 

8  0.433 

Values  of  /  from  equation  (134)   are: 

V=  f= 

400  0.384 

800  0.432 

1600  0.473 

3200  0.512 

It  will  be  noted  that  these  values  of  /  are  larger  than  that 
assumed  in  Art.  112  and  furthermore  that  some  definite  relation 
is  assumed  to  exist  between  the  slip  of  the  belt  and  its  speed. 

Equating  the  two  values  of  /in  (133)  and  (134)  and  neglecting 
the  difference  in  the  constant  terms,  we  have  approximately: 


.0147  (135) 

or  a  total  slip  of  about  6  ft.  per  minute  plus  about  3  per  cent  of 
the  linear  velocity. 

115.  Strength  of  Belting.  —  The  strength  of  belting  varies 
widely  and  only  average  values  can  be  given.  According  to 
experiments  made  by  the  author  good  oak  tanned  belting  has  a 
breaking  strength  per  inch  of  width  as  follows: 


Single 

Double 

Solid  leather  
\Vhere  riveted 

900  Ib. 
600  Ib. 

1,400  Ib. 
1  200  Ib 

Where  laced  

350  Ib. 

Canvas  belting  has  approximately  the  same  strength  as 
leather.  Tests  of  rubber  coated  canvas  belts  4-ply,  8  in.  wide, 
show  a  tensile  strength  of  from  840  Ib.  to  930  Ib.  per  inch  of  width. 


226  MACHINE  DESIGN 

116.  Taylor's  Experiments. — The  experiments  of  Mr.  F.  W. 
Taylor,  as  reported  by  him  in  Trans.  A.  S.  M.  E.,  Vol.  XV,  afford 
the  most  valuable  data  now  available  on  the  performance  of  belts 
in  actual  service. 

These  experiments  were  carried  on  during  a  period  of  nine  years 
at  the  Midvale  Steel  Works.  Some  of  Mr.  Taylor's  conclusions 
are  as  follows: 

1.  Narrow  double  belts  are  more  economical  than  single  ones 
of  a  greater  width. 

2.  All  joints  should  be  spliced  and  cemented. 

3.  The  most  economical  belt  speed  is  from  4000  to  4500  ft. 
per  minute. 

4.  The  working  tension  of  a  double  belt  should  not  exceed  35 
Ib.  per  inch  of  width,  but  the  belt  may  be  first  tightened  to  about 
double  this. 

5.  Belts  should  be  cleansed  and  greased  every  six  months. 

6.  The  best  length  is  from  20  to  25  ft.  between  centers. 

117.  Rules  for  Width  of  Belts.— It  will  be  noticed  that  Mr. 
Taylor  recommends  a  working  tension  only  ^  to  ^  the  breaking 
strength  of  the  belt.     He  justifies  this  by  saying  that  belts  so 
designed  gave  much  less  trouble  from  stoppage  and  repairs  and 
were  consequently  more  economical  than  those  designed  by  the 
ordinary  rules. 

It  must  be  remembered  that  a  belt  which  is  strained  to  an  exces- 
sive tension  will  not  retain  this  tension  long,  but  will  stretch 
until  the  tension  becomes  such  as  the  belt  will  carry  comfortably. 

If  the  belt  is  uncler  size  for  the  required  load  this  will  cause 
slipping  and  necessitate  further  tightening  and  so  on.  There 
will  thus  be  continual  loss  of  time,  so  that  such  a  belt  is  uneco- 
nomical although  theoretically  of  ample  strength. 

In  the  following  formulas  50  Ib.  per  inch  of  width  is  allowed 
for  double  belts  and  30  Ib.  for  single  belts.  These  are  suitable 
values  for  belts  which  are  not  running  continuously.  The 
formulas  may  be  easily  changed  for  other  thicknesses  and  for 
other  values  of  CTV. 

Let  HP  =  horse-power  transmitted 

D  =  diameter  of  driving  pulley  in  inches 
N  =  number  revolutions  per  minute  of  pulley. 


WIDTH  OF  BELTS  227 

The  moment  of  force  transmitted  by  belt  is        * 
RD_CT\wD 
2  =     ~2~ 

TN       CT^DN 
and  HP=  63025  - 


Substituting  the  values  assumed  for  CTl  and  solving  for  w: 

TT  T> 

Single  belts  ^=4200  (137) 


TT  T) 

Double  belts  w  =  2500-^-  (138) 

The  most  convenient  rules  for  belting  are  those  which  give 
the  horse-power  of  a  belt  in  terms  of  the  surface  passing  a  fixed 
point  per  minute. 


In  formula  (136) 
we  will  substitute  the  following: 


IV 

W  =  width  of  belt  in 


7i  DN 
V  =  velocity  in  ft.  per  mm.  =    1 


„„ 

HP-      265& 

Substituting  values  of  C  and  Tt  as  before  and  solving  for 
WV  =  square  feet  per  minute  we  have  approximately: 

Single  belts    WV  =  WHP.  (139) 

Double  belts  WV  =  55#P.  (140) 

118.  Speed  of  Belting.  —  As  in  the  case  of  pulley  rims,  so  in  that 
of  belts  a  certain  amount  of  tension  is  caused  by  the  centrifugal 
force  of  the  belt  as  it  passes  around  the  pulley. 

12wv2 

From  equation  (110)          S  =  - 

i/ 

where  v  =  velocity  in  feet  per  second 

w  =  weight  of  material  per  cubic  inch 
S  —  tensile  stress  per  square  inch. 


228  MACHINE  DESIGN 

To  make  this  formula  more  convenient  for  use  we  will  make 
the  following  changes  in  the  contsants : 

Let  7  =  velocity  of  belt  in  ft.  per  minute  =  60v 
w  =  weight  of  ordinary  belting 

=  .032  lb.  per  cubic  inch 
$!  =  tensile  stress  per  inch  width,  caused  by  centrifugal 

force 
=  about  T3g-  S  for  single  belts. 

Then  v  =  L 


Substituting  these  values  in  (110)  and  solving  for  S1 

V2 
Sl= 1610000 

The  speed  usually  given  as  a  safe  limit  for  ordinary  belts  is 
3000  ft.  per  minute,  but  belts  are  sometimes  run  at  a  speed 
exceeding  6000  ft.  per  minute. 

Substituting  different  values  of  V  in  the  formula  we  have: 

7  =  3000  S,=  5.591b. 

7=4000  S,=  9.941b. 

7  =  5000  ^  =  15.53  lb. 

7  =  6000  51=22.361b. 

The  values  of  S^  for  double  belts  will  be  nearly  twice  those 
given  above.  At  a  speed  of  5000  ft.  per  minute  the  maximum 
•tension  per  inch  of  width  on  a  single  belt  designed  by  formula 
(137),  if  we  call  C  =  .5,  will  be: 

(30x2)+15.=751b. 

giving  a  factor  of  safety  of  eight  or  ten  at  the  splices. 

In  a  similar  manner  we  find  the  maximum  tension  per  inch  of 
width  of  a  double  belt  to  be: 

(50x2)+30  =  1301b. 

and  the  margin  of  safety  about  the  same  as  in  single  belting. 
A  double  belt  is  stiffer  than  a  single  one  and  should  not  be 


ROPE  DRIVES  229 

used  on  pulleys  less  than  1  ft.  in  diameter.     Triple  feelts  can  be 
used  successfully  on  pulleys  over  20  in.  in  diameter. 

119.  Manila  Rope  Transmission. — Ropes  are  sometimes  used 
instead  of  flat  belts  for  transmitting  power  short  distances. 
They  possess  the  following  advantages:  they  are  cheaper  than 
belts  in  first  cost;  they  are  flexible  in  every  direction  and  can 
be  carried  around  corners  readily.  They  have,  however,  the 
disadvantage  of  being  less  efficient  in  transmission  than  leather 
belts  and  less  durable;  they  are  also  somewhat  difficult  to  splice 
or  repair. 

There  are  two  systems  of  rope  driving  in  common  use:  the 
English  and  the  American.     In  the  former  there  are  as  many 
separate    ropes   -as    there    are 
grooves  in  one  pulley,  each  rope 
being    an   endless   loop   always 
running  in  one  groove. 

In  the  American  system  one 
continuous  rope  is  used  passing 
back  and  forth  from  one  groove 
to  another  and  finally  returning 
to  the  starting-point. 

The  advantage  of  the  English     \ 1 

system  consists  in  the  fact  that  FIG.  99. 

one  of  the  ropes  may  fail  with- 
out causing  a  breakdown  of  the  entire  drive,  there  usually 
being  two  or  three  ropes  in  excess  of  the  number  actually  neces- 
sary. On  the  other  hand  the  American  system  has  the  ad- 
vantage of  a  uniform  regulation  of  the  tension  on  all  the  plies 
of  rope.  The  guide  pulley,  which  guides  the  last  slack  turn  of 
rope  back  to  the  starting-point,  is  usually  also  a  tension 
pulley  and  can  be  weighted  to  secure  any  desired  tension.  The 
English  method  is  most  used  for  heavy  drives  from  engines 
to  head  shafts ;  the  American  for  lighter  work  in  distributing  power 
to  the  different  rooms  of  a  factory.  The  grooves  in  the  pulleys 
for  manila  or  cotton  ropes  usually  have  their  sides  inclined  at  an 
angle  of  about  45  degrees,  thus  wedging  the  rope  in  the  groove. 

The  Walker  groove  has  curved  sides  as  shown  in  Fig.  99,  the 
curvature  increasing  toward  the  bottom.  As  the  rope  wears  and 


230  MACHINE  DESIGN 

stretches  it  becomes  smaller  and  sinks  deeper  in  the  groove;  the 
sides  of  the  groove  being  more  oblique  near  the  bottom,  the  older 
rope  is  not  pinched  so  hard  as  the  newer  and  this  tends  to  throw 
more  of  the  work  on  the  latter. 

120.  Strength  of  Manila  Ropes. — The  formulas  for  transmis- 
sion by  ropes  are  similar  to  those  for  belts,  the  values  for  S  and 
W  being  changed.  The  ultimate  tensile  strength  of  manila  and 
hemp  rope  is  about  10,000  Ib.  per  square  inch. 

To  insure  durability  and  efficiency  it  has  been  found  best  in 
practice  to  use  a  large  factor  of  safety.  Prof.  Forrest  R.  Jones 
in  his  book  on  Machine  Design  recommends  a  maximum  tension 
of  200  d2  pounds  where  d  is  the  diameter  of  rope  in  inches.  This 
corresponds  to  a  tensile  stress  of  255  Ib.  per  square  inch  or  a 
factor  of  safety  of  about  40. 

The  value  of  /for  manila  on  metal  is  about  0.12,  but  as  the 
normal  pressure  between  the  two  surfaces  is  increased  by  the 
wedge  action  of  the  rope  in  the  groove  we  shall  have  the  apparent 
value  of/: 

f1=f-s-sm  -=  where 

a  =  angle  of  groove, 
For  a  =  45°  to  30° 


fl  varies  from  0.3  to  0.5  and  these  values  should  be  used  in  for- 
mula (134). 

\l  —  e  *  j  in  this  formula,  for  an  arc  of  contact  of  150  degrees, 
becomes  either  .54  or  .73  according  as/1  is  taken  0.3  or  0.5. 

If  7\  is  assumed  as  250  Ib.  per  square  inch,  the  force  R  trans- 
mitted by  the  rope  varies  from  135  Ib.  to  185  Ib.  per  square  inch 
area  of  rope  section. 

The  following  table  gives  the  horse-power  of  manila  ropes  based 
on  a  maximum  tension  of  255  Ib.  per  square  inch. 


ROPE  DRIVES 


231 


TABLE  LVIII 

Table  of  the  horse-power  of  transmission  rope,  reprinted  from  the  trans- 
actions of  the  American  Society  of  Mechanical  Engineers,  Vol.  XII,  page 
230,  Article  on  "Rope  Driving"  by  C.  W.  Hunt. 

The  working  strain  is  800  Ib.  for  a  2-in.  diameter  rope  and  is  the  same  at 
all  speeds,  due  allowance  having  been  made  for  loss  by  centrifugal  force. 


Diameter 

Speed  of  the  rope  in  feet  per  minute 

Smallest 
diameter 

pulleys, 

1,500 

2,000 

2,500 

3,000 

3,500 

4,000 

4,500 

5,000 

6,000 

7,000 

inches 

1 

3.3 

4.3 

5.2 

5.8 

6.7 

7.2 

7.7 

7.7 

7.1 

4.9 

30 

i 

4.5 

5.9 

7.0 

8.2 

9.1 

9.8 

10.8 

10.8 

9.3 

6.9 

36 

1 

5.8 

7.7 

9.2 

10.7 

11.9 

12.8 

13.6 

13.7 

12.5 

8.8 

42 

li 

9.2 

12.1 

14.3 

16.8 

18.6 

20.0 

21.2 

21.4 

19.5 

13.8 

54 

H 

13.1 

17.4 

20.7 

23.1 

26.8 

28.8 

30.6 

30.8 

28.2 

19.8 

60 

if 

18.0 

23.7 

28.2 

32.8 

36.4 

39.2 

41.5 

41.8 

37.4 

27.6 

72 

2 

23.1 

30.8 

36.8 

42.8 

47.6 

51.2 

54.4 

54.8 

50.0 

35.2 

84 

121.  Cotton  Rope  Transmission. — Cotton  rope  is  more  expen- 
sive than  manila  in  its  first  cost,  but  has  a  greater  efficiency  and 
a  longer  life  than  its  rival.  Instances  are  given  where  cotton 
ropes  have  been  in  continuous  service  for  periods  of  fifteen, 
twenty-five  and  even  thirty  years.  The  rope  of  three  strands 
without  a  core  is  most  flexible  and  durable  as  there  is  good 
contact  between  the  working  strands  and  no  waste  room. 

Mr.  Edward  Kenyon  gives  the  following  values  for  the  power 
which  can  be  safely  transmitted  by  good  three-strand  cotton 
ropes  running  on  pulleys  not  less  than  thirty  times  their  respec- 
tive diameters  (English  system).1  (See  next  page.) 

The  horse-power  at  any  other  speed  will  be  in  proportion  to 
the  speed.  It  will  also  be  noticed  that  the  horse-power  is 
proportional  to  the  square  of  the  diameter  of  the  rope.  Mr. 
Kenyon  gives  figures  for  the  speed  as  high  as  7000  ft.  per  minute, 
and  reports  actual  installations  where  ropes  are  running  success- 
fully at  this  speed.  He  makes  no  allowance  for  centrifugal 
force  and  denies  that  this  has  any  appreciable  effect  on  the 
driving  power  or  the  durability. 

Mr.  Kenyon's  figures  have  reference  only  to  ropes  used  in 
single  plies  as  in  the  English  system. 

1  Am.  Mach.,  July  8,  1909. 


232 


MACHINE  DESIGN 


TABLE  LIX 

HORSE-POWER  OF  COTTON  ROPES — VELOCITY  1000  FT. 
PER  MINUTE 


Diameters  in  inches 

Horse-power 

Rope 

Smallest  pulley 

1 

30 

3.3 

1* 

34 

4.1 

H 

38 

5.1 

If 

41 

6.1 

H 

45 

7.4 

if 

49 

8.6 

if 

53 

10. 

11 

57 

11.5 

2 

60 

13. 

He  calls  especial  attention  to  the  use  of  casing  in  high-speed 
pulleys  to  reduce  the  air  resistance. 

122.  Wire  Rope  Transmission. — Wire  ropes  have  been  used  to 
transmit  power  where  the  distances  were  too  great  for  belting  or 
hemp   rope   transmission.     The  increased  use 
of  electrical  transmission  is  gradually  crowding 
out  this  latter  form  of  rope  driving. 

For  comparatively  short  distances  of  from 
100  to  500  yd.  wire  rope  still  offers  a  cheap  and 
simple  means  of  carrying  power. 

The  pulleys  or  wheels  are  entirely  different 
from  those  used  with  manila  ropes. 

Fig.  100  shows  a  section  of  the  rim  of  such  a 
pulley.  The  rope  does  not  touch  the  sides  of 
the  groove  but  rests  on  a  shallow  depression 
in  a  wooden,  leather  or  rubber  filling  at  the 
bottom.  The  high  side  flanges  prevent  the  rope  from  leaving 
the  pulley  when  swaying  on  account  of  the  high  speed. 

The  pulleys  must  be  large,  usually  about  100  times  the  diam- 


FIG.  100. 


ROPE  DRIVES 


233 


eter  of  rope  used,  and  run  at  comparatively  high  speeds.     The 
ropes  should  not  be  less  than  200  ft.  long  unless  some  form  of 
tightening  pulley  is  used.     Table  LX  is  taken  from  Roebling. 
Long  ropes  should  be  supported  by  idle  pulleys  every  400  ft. 


TABLE  LX 

TRANSMISSION  OF  POWER  BY  WIRE  ROPE 

Showing  necessary  size  and  speed  of  wheels  and  rope  to  obtain  any  desired 
amount  of  power. 


Diameter 
of  wheel 
in  feet 

Number 
of  revolu- 
tions 

Diameter 
of  rope 

Horse- 
power 

Diameter 
of  wheel 
in  feet 

Number 
of  revolu- 
tions 

Diame- 
ter of 
rope 

Horse- 
power 

4 

80 

5-8 

3.3 

10 

80 

11-16 

58.4 

100 

5-8 

4.1 

100 

11-16 

73. 

120 

5-8 

5. 

120 

11-16 

87.6 

140 

5-8 

5.8 

140 

11-16 

102.2 

5 

80 

7-16 

6.9 

11 

80 

11-16 

75.5 

100 

7-16 

8.6 

100 

11-16 

94.4 

120 

7-16 

10.3 

120     , 

11-16 

113.3 

140 

7-16 

12.1 

140 

11-16 

132.1 

6 

80 

1-2 

10.7 

12 

80 

3-4 

99.3 

100 

1-2 

13.4 

100 

3-4 

124.1 

120 

1-2 

16.1 

120 

3-4. 

148.9 

140 

1-2 

18.7 

140 

3-4 

173.7 

7 

80 

9-16 

16.9 

13 

80 

3-4 

122.6 

100 

9-16 

21.1 

100 

3-4 

153.2 

120 

9-16 

25.3 

120 

3-4 

183.9 

8 

80 

5-8 

22. 

14 

80 

7-8 

148. 

100 

5-8 

27.5 

100 

7-8 

185. 

120 

5-8 

33.0 

120 

7-8 

222. 

9 

80 

5-8 

41.5 

15 

80 

7-8 

217. 

100 

5-8 

51.9 

100 

7-8 

259. 

120 

5-8 

62.2 

120 

7-8 

300. 

PROBLEMS 

1.  Design  a  main  driving  belt  to  transmit  200  horse-power  from  a  belt 
wheel  18  ft.  in  diameter  and  making  80  revolutions  per  minute.     The  belt 
to  be  double  leather  without  rivets. 

2.  Investigate  driving  belt  on  an  engine  and  calculate  the  horse-power 
it  is  capable  of  transmitting  economically. 


234  MACHINE  DESIGN 

3.  Calculate  the  total  maximum  tension  per  inch  of  width  due  to  load  and 
to  centrifugal  force  of  the  driving  belt  on  the  motor  used  for  driving  machine 
shop,  assuming  the  maximum  load  to  be  10  horse-power. 

4.  Design  a  manila  rope  drive,  English  system,  to  transmit  400  horse- 
power, the  wheel  on  the  engine  being  20  ft.  in  diameter  and  making  60  revo- 
lutions per  minute.     Use  Hunt's  table  and  then  check  by  calculating  the 
centrifugal  tension  and  the  total  maximum  tension  on  each  rope.     Assume 

v2 
*^=  en  wnere  v=  fee"k  Per  second. 

oU 

5.  Design  a  wire  rope  transmission  to  carry  150  horse-power  a  distance  of 
one-quarter  mile  using  two  ropes.     Determine  working  and  maximum  ten- 
sion on  rope,  length  of  rope,  diameter  and  speed  of  pulleys  and  number  of 
supporting  pulleys. 

REFERENCES 

Manufacture  of  Belting.     Power,  Feb.,  1903. 

Steel  Belts.     Am.  Mach.,  Jan.  24,  1908;  Eng.  News,  Oct.  14,  1909. 

Belts  vs.  Ropes.     Power,  Dec.  1,  1908. 

Lewis'  Experiments  on  Belts.     Tr.  A.  S.  M.  E.,  Vol.  VII,  p.  549. 

Transmission  of  Power  by  Belting.     Tr.  A.  S.  M.  E.,  Vol.  XX,  p.  466; 

Vol.  XXXI,  p.  29. 

Various  Systems  of  Rope  Transmission.     Am.  Mack.,  July  8,  1909. 
Rope  Driving.     (Hunt.)     Tr.  A.  S.  M.  E.,  Vol.  XII,  p.  230. 
Working  Load  for  Ropes.     Tr.  A.  S.  M.  E.,  Vol.  XXIII,  p.  125. 


CHAPTER  XIII 

• 

DESIGN  OF  TOGGLE-JOINT  PRESS 

123.  Introductory  Statement. — In  discussing  the  subject  of 
Machine  Design  much  time  may  be  saved  by  assuming  some  simple 
machine  and  illustrating  methods  in  design  by  a  fairly  complete 
analysis  of  all  the  important  theoretical  calculations.  Such  a 
layout  at  once  gives  the  scope  of  the  work  and  protects  the  beginner 
from  so  much  "  working  in  the  dark.'7  An  assignment  may  then 
be  made,  differing  in  a  lesser  or  greater  degree  from  the  illustrated 
design  and  a  complete  analysis  required  of  all  parts  of  the  machine. 
After  the  student's  experience  with  the  first  design  he  will  need 
the  second  one  developed  less  elaborately  and  possibly  the  third 
one  not  at  all. 

Design  No.  1  is  meant  especially  to  cover  static  forces,  i.e., 
simple  applications  of  members  in  tension,  compression,  flexure 
and  shear.  A  good  illustration  of  this  is  the  toggle-joint  press. 
Machines  of  this  class  are  sometimes  used  in  forming  thin  sheets  of 
copper  and  brass  into  articles  for  ornamental  purposes,  conse- 
quently it  is  a  useful  tool.  Plates  C-l,  C-2,  and  C-3  show  a 
design  of  a  small  machine  and  are  inserted  to  give  an  idea  as  to  the 
arrangement  of  the  drawings.  The  design  was  worked  up  on 
three  12  in.  Xl8  in.  sheets;  two  of  details  and  one  assembly  view. 

It  is  urged  that  the  designer  regard  these  sheets  merely  as 
illustrative  of  a  good  drawing  room  job  and  that,  from  the  stand- 
point of  ideas,  he  will  cultivate  originality  and  make  a  design  as 
nearly  independent  as  possible. 

Alternative  Designs  will  be  found  at  the  end  of  this  chapter. 
These  may  be  substituted  for  the  regular  designs  if  preferred. 

In  designing  a  complete  machine  each  part  should  be  worked 
up  as  an  independent  unit  but  with  all  available  information  as 
to  its  relation  to  the  other  parts  composing  the  machine.  Be- 
fore attempting  to  develop  any  individual  part,  the  designer  should 
have  a  good  idea  of  what  the  machine  looks  like.  Free-hand 

235 


236  MACHINE  DESIGN 

sketching  should  be  insisted  upon.  These  sketches  when  satis- 
factory should  become  a  part  of  the  report  and  be  handed  in  with 
the  finished  drawings.  Calculate  by  rational  formula  every  part 
that  will  admit  of  such  treatment.  Where  the  conditions  of 
stresses  are  not  well  known  apply  empirical  rules  and  the  best 
approximations  possible.  In  any  case  the  judgment  of  the 
designer  must  be  used  to  modify  and  check  even  the  best  rational 
or  empirical  rules.  Theoretical  deductions  should  not  be  mini- 
mized but  good  j  udgment  should  be  emphasized.  All  calculations 
should  be  saved  until  the  entire  design  is  finished  and  these  should 
be  kept  in  the  exact  order  of  development.  Sometimes  a  part  that 
is  at  first  considered  wrong  may  later  be  found  to  be  correct  and 
recalculation  is  avoided.  Occasionally  it  is  necessary  to  review 
part  of  the  calculations  to  prove  some  part  of  the  design.  Where 
the  theoretical  work  is  neatly  made  and  logically  arranged  this 
may  be  done  without  much  loss  of  time.  In  the  analysis  of  the 
forces  and  the  calculations  of  the  various  parts  a  high  degree 
of  refinement  should  be  aimed  at  for  the  sake  of  showing  how  prin- 
ciples are  applied,  even  though  the  illustrative  piece  may  not 
demand  a  very  thorough  analysis.  The  object  sought  is  not  so 
much  that  a  machine  be  designed  by  the  student  as  that  he 
be  fortified  with  the  ability  to  analyze  a  problem  and  that  he  be 
able  to  apply  to  it  the  correct  principles  of  design. 

124.  Drawings. — The  following  dimensions  are  suggested  for 
the  cutting  sizes  of  the  sheets.  The  designer  is  at  liberty  to  make 
his  own  selection  from  these  sizes.  It  is  suggested,  however, 
that  the  sheets  be  taken  as  small  as  will  admit  of  a  clear  and 
distinct  set  of  drawings. 

24  in.X36  in.— Size  A 

18  in.X24  in.— Size  B 

12  in.XlS  in.— Size  C 

9  in.Xl2  in.— Size  D 

Scale. — Any  scale  may  be  taken  which  will  show  clearly  all  the 
details  and  give  a  good  arrangement  on  the  sheet.  Details  may 
have  different  scales  on  the  same  sheet  if  so  desired.  When 
this  is  done  each  detail  should  have  the  scale  given. 


'IT 


PLATE  C  1 


50"- 


Note:  See  details  on  drawings 
C— 2  and  C— 3. 


TOGGLE  JOINT  PRESS 
ASSEMBLY 

Scale  — 

Fin-due  Uoiv«rsitV      L 
by 
by 


INSTRUCTIONS  CONCERNING  DRAWINGS  237 

Border  Line. — A  margin  of  J  in.  should  be  left  between  the 
border  line  and  the  edge  of  the  finished  sheet  on  the  top,  bottom 
and  right  end,  and  J  in.  on  the  left  end  to  allow  for  punching  and 
fastening. 

Name  Plate. — Make  the  name  plate  or  title  at  the  lower  right- 
hand  corner  to  cover  a  space  about  2J  in.  X3^  in.  If  any  other 
standard  corner  is  preferred  other  dimensions  may  be  sub- 
stituted. No  border  line  need  be  drawn  around  the  name  plate. 
It  would  be  well  for  each  designer  to  make  a  standard  corner 
plate  to  be  used  below  the  various  tracings  when  working  up  this 
part. 

All  drawings  will  be  carefully  worked  up  in  pencil  and  turned 
in  to  the  instructor.  The  instructor  will  give  them  to  another 
designer  who  will  be  responsible  for  the  checking.  Checking 
will  be  done  in  the  form  of  notes  on  a  separate  paper  and  attached 
to  the  drawings.  These  notes  and  drawings  will  then  be  returned 
to  the  designer  for  approval  and  corrections.  When  the  designer 
traces  his  drawings,  or  such  part  of  them  as  may  be  selected  by 
the  instructor,  he  will  obtain  the  signature  of  the  checker  to 
them  and  submit  the  same  with  the  checker's  notes  to  the 
instructor  for  approval. 

Each  designer  should  have  experience  not  only  in  planning  and 
executing  well  his  own  designs,  but  he  should  take  up  designs  of 
other  men  and  offer  suggestions  and  criticisms  upon  their  work. 
One  way  to  obtain  this  experience  has  been  suggested  above. 

In  checking  up  the  work  of  another  man  the  following  points 
should  be  observed: 

(1)  General  appearance  of  the  design  relative  to  workman- 
ship and  execution,   arrangement  of  drawings,   notes,   dimen- 
sions, etc. 

(2)  General   design   relative    to     proportion,     strength   and 
arrangement  of  parts.     This  is  to  be  merely  the  checker's   im- 
pression  and   need   not   require  the   checking  of  the '  original 
calculations. 

No  drawing  should  be  retained  longer  than  one  exercise  and 
at  the  completion  of  the  checking  should  be  returned  to  the 
designer.  It  is  estimated  that  any  set  of  drawings  may  be 
checked  in  this  way  within  two  hours'  time.  No  notes  or  marks 
will  be  made  on  the  drawings  but  special  paper  will  be  provided 


238  MACHINE  DESIGN 

for  this  purpose.  In  looking  over  the  drawings  finally,  the  in- 
structor will  give  credit  to  the  work  of  the  checker  as  well  as  to 
that  of  the  designer. 

In  all  this  work  some  standard  text  on  mechanical  drawing 
should  be  adopted  as  reference  concerning  arrangement  of  views, 
sectioning,  cross-hatching,  lettering,  and  the  like. 

Every  dimension  should  be  clearly  shown  so  that  no  measure- 
ments need  be  taken  by  scale  from  the  drawing. 

All  dimensions  should  be  given  in  round  vertical  figures, 
heavy  enough  to,  print  well.  No  diagonal-barred  fractions,  thin 
or  doubtful  figures  should  be  accepted. 

All  dimensions  should  be  given 
in  inches. 

All  dimension  lines  should  be 
made  as  light  as  will  insure  good 
printing  and  should  have  a  central 


space  for  figures. 


36' 

All  dimensions  should  read  in 
FIG.  101.  ,,      -,.      ,.         ,.  ,, 

the  direction  01  the  arrows. 

Avoid  crowding  the  dimensions  to  the  center  of  any  detail.  A 
much  better  way  is  by  the  use  of  projected  lines  as  shown  by 
Fig.  101. 

All  detailed  pieces  should  be  accompanied  by  a  shop  note 
or  call  as  "C.  I.  One  wanted";  "  M.  S.  Two  wanted";  "Finish 
all  over;"  "Turned  for  a  shrinking  fit;"  etc.,  etc. 

The  following  abbreviations  will  be  considered  satisfactory  in 
these  calls: 

C.  S Cast  steel.  f Finish  (see  sheets  of  details). 

C.  I Cast  iron.  B.  b.  t..  Babbitt  metal. 

W.  I Wrought  iron.  D Diameter. 

M.  S Machine  steel.  R Radius. 

125.  Calculations. — Each  designer  is  expected  to  draw  up  a 
report  in  parallel  with  the  design.  This  report  should  contain 
such  free-hand  sketches  as  relate  to  the  calculations,  also  a  full 
report  of  the  calculated  sizes  and  accepted  sizes  of  the  different 
parts  of  the  design,  and  be  submitted  in  a  manila  cover  with  the 
finished  tracings  and  drawings. 


n^/ 


'54.^.,.  . 


-llil 


PLATE  C  2 


•4-  hole.* 


-» 


i"x  la"  OQp  3CTOW 


*** 


uiCO 


Drill  i"  For  Pi?  'V  8"  MS 


n 


frl 


Die  Head    C.I. 


C.5.  Per 
.'sc.    2  react. 


-8' 


Drill  y  For  Pi7 


w  -i 

i  -2 

'«  r- 

i  V 

.r  o" 

Q.  £- 

&  -6 


1° 


'?  ?Utr"4"-t 
fe 


•^rrrr 
T~  'i 


M.S. 


t    S  Teq.'d 


LJ    1 


C.I. 


M-* 

^ 


^ 


• 

4 

— J  1    "® 
i¥ 


P!' 

_T  — 


T 
i 


Not*-.- 


TDEELE  JOINT  PRESS 
DETAILS 

Scale. i^Q?d  e  size 
RiTdue  U^versity          LaFayette.  \i 
Drctwrp 


. 


TOGGLE-JOINT  DETAILS 


239 


Design  No.  1 

TOGGLE-JOINT  PRESS 

126.  Analysis  of  the  Forces  Involved. — Referring  to  Plate  C-l, 
it  will  be  seen  that  the  acting  forces  can  be  represented  in  the 
following  force  diagram,  with  the  direction  of  the  forces  repre- 
sented by  arrows. 


FIG.  102. 

Each  designer  will  be  given  a  value  for  W,  I,  V  and  0.  In  all 
the  designs  <£  may  be  taken  at  10°,  assuming  that  the  maximum 
load  will  be  carried  at  this  position  and  that  the  lever  arm  will 
then  be  horizontal. 

In  the  assignments  for  a  number  of  designs  the  range  of  values 
may  be  as  follows: 

TF  =  200,  300,  400 1000  Ib. 

1  =  4,  4.5,  5,  5.5 10  ft. 

V  =for  large  sizes,  6,  8,  10 12  in. 

for  small  sizes  6,  6.5,  7 8  in. 

Selecting  for  our  analysis  the  following  values:  TF  =  100  Ib.; 
1  =  5  ft.  3  in.;  I' =  7  in.;  and  <£  =  10  degrees,  we  have  from  the 
force  diagram 

Wl      100X63 


W  = 


V 


7 
100X56 


=  9001b. 


=  8001b. 


W 


900 


=  2591.4  Ib. 


2  sin.  cf)     .3473 
W3  =  W^  cos  <f>  =  2591.4  X. 98481=  2552  Ib. 


240 


MACHINE  DESIGN 


127.  Lever. — The  formula  for  calculating  beams  in  flexure, 
Art.  4,  is  M  =  SZ,  where  M=  bending  moment  in  pounds- 
inches,  S  =  working  fiber  stress  in  pounds  and  Z  =  resistance  of 
the  section  or  modulus.  In  any  section  of  the  lever  transversely 
across  the  axis  let  b  and  h  be  the  breadth  and  the  height  of  the 
section  respectively.  The  designer  must  here  decide  if  the  beam 
is  to  have  parallel  sides,  in  which  case  b  will  be  constant  for  all 
sections,  or  taper  sides,  in  which  case  a  certain  ratio  of  b  to  h 
would  be  used.  The  best  way  to  decide  which  to  use  is  to  find 
the  size  of  the  sections  at  two  critical  points  as  g  and  c,  Fig. 
103  (c  is  the  fulcrum  and  g  is  any  point  near  the  handle),  for 
each  case  and  select  between  them.  Assuming  iS  — 8000  for 


•22- 


m 


FIG.    103. 


wrought  iron  or  mild  steel,  6  =  1,  and  disregarding  the  hole  at  c, 
which  has  little  effect  since  the  fiber  stress  of  any  section  ap- 
proaches zero  at  the  center,  our  formula  M  =  SZ  gives 
(section  at  g)  100 X   6  =  8000x1  X/>2^6;  h=   .67  in. 
(section  at  c)  100X56  =  8000x1  Xh2 -=-6;  h  =  2.05  in. 
This  beam  would  have  a  better  shape  and  would  also  be  lighter 
if  the  thickness  be  reduced  below  1  in.,  say  to  .75  in.    With  this 
value  the  formula  gives 

(At  g)  100X   6  =  8000X.75X&2^6;  h=   .77  in. 
(At  c)  100 X 56 - 8000 X. 75 X h2 -6;  h -2.37  in. 
These  values  give  a  well-shaped  beam,  having  a  section  .75  in. 
X.77  in.  at  g  and  .75  in.  X  2.37  in.  at  c. 

On  the  other  hand,  suppose  a  ratio  of  6  to  h  =  \,  to  be  desired, 
the  problem  becomes 

(At  g)  100X6  =  8000  ft3 -=-24;  h  =  1.22  in. 
and  6  =  1.22 -=-4  =  .3  in. 


L_ 


. 


i  -^—H 


«..-«-,«-  «*«. 


PLATE  C  3 


2." 


! 


Alsorec^'eJ  -  O7O  ^.  bolt,  SQ    '°?*J« 
all  over,    tfireoded    I"  Wit^two 


TDGGLE3  JOINT  PRESS 
DETAILS 

Scale,    £  size. 


locd-i}utft  .  P<ac«d  oTr  to  a  tKi 

r* 


l_ 


•*•  t 

•V— -T       g,-p      - 


.,•-4— -_:- +i-.,  _.).._• 

:r.>«fc±s:&<    r    ; 


TOGGLE-JOINT  DETAILS  '  •  241 

(At  c)  100X56  =  8000  h3-r- 24;  h  =  2.56  in. 
and  6  =  2.56 -r-4  =  . 64  in. 
section  at  g=   .3  in.  X  1.22  in. 
section  at  c  =  .64  in.  X  2. 56  in. 

The  above  gives  the  method  of  determining  the  size  of  the 
section  at  any  point  of  the  beam.  Sections  should  be  taken  at 
regular  intervals  of  length  and  a  diagram  plotted  from  the  results. 
One  section  only  need  be  taken  between  a  and  c,  say  at  o  midway 
between.  This  diagram  when  completed  will  show  the  beam  to 
take  the  form  of  a  curve  similar  to  Fig.  104.  It  may  be  found 
convenient,  however,  to  approximate  this  curve  with  a  straight 
line  as  x  y.  This  would  be  satisfactory  for  strength  and  would 
be  more  easily  constructed. 

It  will  be  noticed  that  the  bending  moment  becomes  zero  at  the 
points  a  and  p  where  the  loads  are  applied.  This  would  theoret- 
ically give  no  size  to  the  handles  and  make  it  impossible  of 


FIG.  104. 

construction.  Some  satisfactory  design  of  handle  or  hub  must 
be  made  at  these  points  with  sufficient  size  to  carry  the  pins  or 
bolts,  each  hub  to  have  the  sides  and  edges  of  the  beam  filleted 
into  it  in  a  neat  manner.  See  Plate  C-3.  A  handle  can  be 
placed  at  p  for  all  loads  of  300  Ib.  or  less  and  a  drilled  hub  for 
larger  loads  so  that  a  small  air  or  steam  cylinder  can  be  attached. 
A  similar  hub  will  be  added  at  a,  for  connection  to  the  post  at 
the  rear. 

128. — The  following  shapes  may  be  found  useful  in  designing 
the  lever. 

Shapes  at  p. — The  size  and  shape  of  the  handle  or  hub  at  this 
end  will  be  largely  a  question  of  neatness,  since  the  load  carried  is 
very  small.  The  pin,  if  one  is  used,  may  be  calculated  for 


242 


MACHINE  DESIGN 


\ 


FIG.  106. 


TOGGLE-JOINT  DETAILS 


243 


double  shear  to  get  the  minimum  size  allowable,  but  this  size 
will  probably  be  so  small  that  it  will  be  necessary  to  increase  the 
size  of  both  pin  and  hub  to  add  symmetry  to  the  design.  Such 
points  as  this  call  for  special  investigation.  Any  piece  of  a 
machine  may  be  made  extra  strong,  if  necessary  to  harmonize 
with  the  other  parts  of  the  machine,  but  the  reverse  is  not  the  case. 
Construction  of  the  Joint  at  a. — Referring  to  Fig.  106,  shapes 
A  and  B  would  be  preferred.  In  most  cases  the  standard  would 
be  made  of  cast  iron  and  could  easily  be  cored  out  to  fit  over 
the  lever  arm  end  rather  than  to  fit  the  arm  end  over  the  stand- 
ard as  at  C.  The  only  calcu- 
lations necessary  for  this  end 
of  the  lever,  besides  figuring 
the  pin,  are  those  that  deter- 
mine the  diameter  of  the  hub 
and  the  length  of  the  hub. 
It  is  reasonable  to  assume 
that  the  diameter  of  this  hub 
should  be  made  equal  to  the 
diameter  of  the  cast  hub  of 
the  standard.  To  illustrate:  at  a 


FIG.  107. 
a  tensional  force  of  W"  is 


acting  upward  and  this  force  is  resisted  by  four  cast  iron  areas 
on  the  section,  RS,  equal  in  total  area  to  R'S',  of  the  standard 
(Fig.  107). 

These  four  areas  are  produced  by  passing  a  plane  through  the 
standard  along  the  line  RS.  Each  area  is  equal  to  bh  and  should 
be  figured  for  cast  iron  in  direct  tension  by  the  formula  W  =  SA. 
In  making  this  calculation  the  ratio  of  b  to  h  may  be  assumed. 
Having  figured  the  pin  for  double  shear  by  the  formula  W"  =  2SA. 
find  the  diameter  of  the  pin  and  add  to  it  2h,  which  will  give  the 
diameter  of  the  cast  hub  and  consequently  the  diameter  of  the 
lever  end.  If  S  for  shear  in  wrought  iron  be  taken  at  5000  Ib. 
per  square  inch,  the  diameter  of  the  pin  will  be  .33  in.  or,  say  f  in. 
If  S  for  tension  in  cast  iron  be  taken  at  1500  Ib.  per  square  inch, 
the  area  bh  will  be  .133  square  inch,  from  which,  if  b  be  taken  at 
J  in.,  h  becomes  .53  in.  This  would  make  the  diameter  of  the 
hubs  at  a,  If  in. 

It  will  be  next  in  order  to  find  the  length  of  the  hub  at  the  lever 
end,  also  the  corresponding  values  of  the  standard  top.  These 


244  MACHINE  DESIGN 

are  determined  largely  from  the  crushing  strength  of  the  pin. 
First  examine  b  of  the  standard  to  see  if  the  assumed  J  in.  is 
sufficient.  The  part  of  the  pin  in  the  casting  and  the«part  in  the 
lever  are  both  subjected  to  a  crushing  force.  The  resistance  of 
the  pin  to  crushing  is  in  proportion  to  the  projected  area  of  that 
part  of  the  pin  involved. 

In  Fig.  108  let  the  pin  be  cut  by  a  horizontal  plane  through  its 
diameter  1,  6,  7,  4,  corresponding  to  the  plane  along  RS  of  the 
standard.  1,  2,  3,  4  and  5,  6,  7,  8  are  the  projections  of  the  parts 


FIG.  108. 

included  within  the  arms  of  the  standard  and  2,  5,  8,  3  is  the  pro- 
jection of  that  part  included  within  the  lever  end.  The  diameter 
of  the  pin  has  previously  been  figured  to  resist  shearing  along 
the  two  planes  2,  3  and  5,  8.  Now  it  is  necessary  to  find  the 
length  1,  2  and  5,  6  such  that  these  parts  will  be  safe  from  crush- 
ing. For  the  part  in  the  casting,  26d  =  2Xi  Xf  =  A  sq.  in.  = 
areas  1,  2,  3,  4  +  5,  6,  7,  8.  If  now  the  factor  of  safety  for  the 
wroughtiron  pin  be  taken  so  that  5000  is  a  safe  value  for  shear, 
Ss,  the  pin  will  sustain  W  =  /S8A  =  1\X5000  =  938  Ib.  safely. 
This  we  find  is  greater  than  the  load  W"  actually  pulling  on  the 
standard  so  that  part  of  the  pin  within  the  castiron  standard  is 
safe.  If  it  had  been  found  that  2bd  was  so  small  that  the  load 
it  was  capable  of  sustaining  safely  before  crushing  was  less  than 
the  load  applied,  then  either  b  or  d  or  both  would  be  increased. 
If  d  were  increased  without  changing  6  then  the  hub  diameter 
would  be  increased  this  amount  above  the  calculated  size  of  If  in., 
but  if  b  were  increased,  the  areas  bh  would  be  stronger  than  the 
calculated  value  and  h  could  be  reduced  accordingly,  if  it  were 
considered  necessary. 


TOGGLE-JOINT  DETAILS 


245 


By  the  same  line  of  reasoning  the  length  of  the  pin  within  the 
lever  would  determine  the  mimimum  length  of  the  lever  hub  to 
resist  crushing.  This  would  be  2b  =  %  in.  From  inspection  it  is 
readily  seen  that  the  thickness  at  a  must  be  necessarily  increased 
to  that  of  the  lever  section.  This  at  c  is  .64  in. 

In  every  fastening  of 
this  kind,  investigation 
may  be  made  for  shearing 
of  the  pin,  the  strength  of 
the  sections  around  the 
pin,  and  the  crushing  of 
the  pin,  within  both  lever 
and  standard. 


o 


129. — In  calculating  the 
size  of  the  section  at  c  the 
hole  was  not  considered.  ( 
The  error  introduced  by 
this  is  very  slight  and  in 
most  cases  may  be  neg- 
lected. The  fiber  stress 
in  the  cross-section  of  the 
arm  varies  from  zero  near 
the  center  to  a  maximum 
at  the  edge  as  shown  in 
diagram  B,  Fig.  109,  where 
by  proportion  we  can 
readily  obtain  the  relative 
resistance  offered  by  the 
metal  at  the  center  as  com- 
pared to  that  at  the  edge  of 
the  section.  The  loss  at 
the  center  is  more  than 
taken  up  by  the  addition 
of  a  fraction  of  an  inch  at  the  edge  or  a  very  small  boss  around 
the  hole.  If  the  hole  in  any  case  should  be  large,  a  modulus 
could  be  selected  for  this  hollow  section,  and  the  exact  sizes 
obtained. 

The  pin  would  be  calculated  in  double  shear — as  at  a. 


FIG.  109. 


246 


MACHINE  DESIGN 


The  size  of  the  boss,  if  any  be  added,  is  largely  optional  and  is 
put  on  for  finish. 

130.  Screw  Fastening  for  Standard. — In  deciding  upon  the 
kind  of  fastening  between  the  standard  and  the  bed,  it  would  be 
well  to  first  examine  it  regarding  the  turning  moments  about  a, 
Fig.  110,  where  W"b  +  W3h3-  W'Jb'  =  Wxl'  +  W yl" .  Assume 
b  =  b'=3  in.,  h3  =  2  in.,  Z'  =  5  in.  and  Z"  =  l  in.  then  with  TP"  =  800, 
TF3  =  2552,  and  TF'4  =  450  lb.  We  have  5  TFz  +  TFy  =  6154  inch- 
pounds. 

If  Wx  =  Wy  then  6  Wx  =  6154  or  Wx  =  1026  lb.  This  is  equiva- 
lent to  a  ^-in.  bolt.  Suppose  Wy, 
because  of  its  location,  to  be  of  little 
value  in  resisting  turning  about  a, 
then  5  TFX  =  6154  and  17*  =  1231  lb. 
=  approximately  iVm-  bolt.  If 
more  than  one  bolt  is  used  along  the 
line  Wx  or  Wy  then  the  total  bolt 
area  at  the  root  of  the  threads  may 
be  the  equivalent  of  that  given 
above. 

Next  examine  the  joint  for  a  sum-^ 
mation  of  all  vertical  forces. 

W"  —  W'4  =  force  holding  stand- 
ard to  bed  =  Wx  +  Wy.     IiWx=  Wy 
then,  2  Wx  =  800  —  450  =  350  lb.  and 
FIG.  110.  Wx  =  175  lb. 

Since  this  force  is  less  than  that 
obtained  by  moments  it  need  not  be  considered. 

Next  examine  the  joint  for  a  summation  of  the  horizontal 
forces.  In  this  the  force  W3  tends  to  shear  the  bolts  off  in  a 
plane  with  the  top  of  the  bed.  It  also  acts  upon  the  flanges  to 
shear  the  casting  inside  the  bolt  holes.  Considering  the  bolts 
first 

W3  =  SA .     If  we  take  S  =  5000,  then 
2552  =  5000A;  ,4  =  .51  sq.  in.  of  bolt  area. 

If  the  bolt  shears  at  the  root  of  the  thread,  as  would  be  the  case 
with  a  cap  screw,  we  need  at  least  four  f-in.  cap  screws. 


TOGGLE-JOINT  DETAILS 


247 


In  the  second  case,  if  the  flange  is,  say  6  in.  long  and  t  in. 
thick,  we  have  for  the  two  sides  2552  =  2X6  tS.  Let  £  =  1500 
for  cast  iron  and  find  t  =  approximately  .15  in. 

This  would,  of  course,  be  made  thicker,  say  J  to  J  in.,  for  the 
appearance  and  good  proportion  of  the  casting. 

In  the  above  discussion  of  the  standard  fastening,  the  part 
most  liable  to  fail  would  be  the  shearing  of  the  bolts.  This 
might  not  be  true  in  every  case;  for  example,  if  h3  were  very  great 
when  compared  to  I' ,  the  failure  of  the  joint  would  probably  be 
by  moments  about  a.  The  above  calculations  would  be  modified, 
also  by  the  arrangement  of  the  bolts  or  cap  screws. 

It  is  well  in  every  case  to  examine  a  joint  from  all  standpoints 
and  design  for  the  greatest  requirement. 

131.  Standard. — The  design  of  the  standard  would  depend 
largely  upon  the  magnitude  of  the  force  to  be  resisted.  In  the 
smaller  machines  it  would  undoubtedly  be  made  of  cast  iron  and 


FIG.  111. 

as  such  the  upper  end  would  be  as  shown  in  the  preceding 
paragraph.  In  the  larger  machines  the  standard  would  be 
made  of  wrought  iron  or  steel  plates,  in  which  case  the  sizes  of 
the  standard  and  lever  end  would  be  calculated  from  different 
values  of  S  than  those  used  for  cast  iron. 

The  cross-section  of  the  body  of  a  cast  iron  standard  may  be 
shaped  as  in  Fig.  111.  Assuming  the  areas  to  be  equal,  the 
strongest  section  to  resist  any  bending  action  that  may  come 
upon  it,  is  D. 

The  lower  end  of  the  standard  would  be  planned  to  receive  the 
rod  W2,  and  would  have  a  flange  for  fastening  to  the  top  of  the 
bed.  Fig.  112  shows  some  of  the  shapes  that  may  be  used. 

The  pin  at  the  base  is  figured  for  double  shear  by  the  formula 


248 


MACHINE  DESIGN 


NOTE. — When  the  constant  2  is  used  in  the  formula  for  double 
shear  the  result  is  the  single  cross-section  of  the  piece.  When 
this  constant  is  omitted  as  in  W  =  SA  the  result  is  the  combined 
cross-sectional  area. 


C 


FIG.  112. 


132.  Toggle. — There  are  three  ways  in  which  the  toggle  may 
fail  at  the  central  joint:  by  shearing  the  pin,  by  bending  the  pin 
and  by  crushing  the  pin.  In  Fig.  113,  B  shows  a  very  simple 
arrangement  of  this  point.  To  obtain  the  size  of  the  pin  in  this 
case  to  resist  shearing 


Wi 


Wi 

2 


FIG.  113. 

W,  =  W2  =  2SA .     IiS  =  5000,  then 
2591.4  =  2X50004.;  A  =  .26  sq.  in.  and  d  =  .5S  say  f  in. 
It  is  readily  seen  that  the  pin  would  be  found  to  be  the  same 

W 

size  if  the  load  -~  were  figured  for  single  shear  as  if  W±  were 

figured  for  double  shear. 

To  obtain  the  size  of  the  pin  to  resist  bending  assume  some 


TOGGLE-JOINT  DETAILS 


249 


length  of  pin  between  the  outer  forces  -—-,  as  2  in.  or  3  in.,  and 

solve  by  the  formula  W'l  +  8  =  SZ.  See  Art.  4.  There  might 
be  a  question  raised  here  concerning  the  proper  formula  to  use 
for  the  bending  moment,  i.e.,  fixed  ends  or  free  ends.  With  the 
two  ends  of  the  pin  held  somewhat  rigidly  between  the  two  sets 
of  resisting  forces,  it  is  in  about  the  same  condition  as  a  beam 


FIG.  114. 

fixed  at  the  ends  and  loaded  at  the  middle.  If  $  =  8000  and 
1  =  2,  then  900X2-8  =  8000X7rd3-^-32;  d  =  .65  say  \\  in. 

The  toggle  action  on  the  pin  at  the  center  requires  that  the 
smaller  force  W  should  come  at  the  center  of  the  length  of  the 
pin  as  shown  in  A  and  B,  Fig.  113.  If  the  heavier  force  W1  or  W2 
acts  at  the  center  of  the  pin  it  would  cause  an  unnecessary  bend- 
ing as  shown  in  C,  and  would  require  too  large  a  pin  to  resist 
this  stress. 

Fig.  114  shows  other  methods  of  designing  the  toggle. 

Concerning  the  crushing  of  the  pin  see  Art.  128. 


250 


MACHINE  DESIGN 


133.  Fig.  115  gives  some  shapes  of  toggle  members.  A,  B,  C, 
and  D  are  usual  shapes  of  the  horizontal  members.  A  and  B 
have  split  ends  and  are  necessarily  hard  to  forge  and  machine. 
C  is  the  simplest  form.  This  form  is  sometimes  modified  by 
adding  bosses  to  one  or  both  sides  as  shown  in  D.  The  vertical 
member  may  be  constructed  solid  as  at  E  or  adjustable  as  at  F. 


FIG.  115. 


134.  Die  Heads. — The  sliding  head  receives  the  thrust  W1 
from  the  toggle  and  moves  along  the  top  of  the  bed  toward  or 
from  the  stationary  head.  It  must  be  a  good  fit  to  the  bed  top 
having  a  free  sliding  contact  but  no  side  motion.  The  stationary 
head  must  be  planned  for  longitudinal  adjustment  and  for  fasten- 


TOGGLE-JOINT  DETAILS 


251 


ing  rigidly  to  the  bed  top  when  desired.  Suggestions  for  attach- 
ing these  heads  are  shown  in  Fig.  116.  Rectangular  and  V-- 
shaped ways  are  used,  some  having  adjusting  gibs  and  some 
plain.  A  is  the  simplest  form  and  may  be  grooved  from  the  solid 
or  held  down  by  plates.  In  such  a  design  the  overlap  below  the 
top  of  the  bed  should  be  made  sufficiently  strong  to  resist  the 
turning  action  from  Ws.  B  and  C  show  the  application  of  gibs 
between  the  sliding  head  and  the  bed  to  take  up  side  slack.  In 
some  classes  of  machines  gib  arrangements  are  essential.  If, 


FIG.  116. 

however,  heavy  side  thrusts  are  involved  the  form  C  is  question- 
able unless  made  very  heavy  and  strong.  With  the  bed  planed 
to  an  angle  as  at  C  and  D,  the  latter  would  be  considered  the 
stronger. 

Sliding  Head. — Since  the  sliding  head  cannot  be  rigidly  fastened 
to  the  bed,  it  must  be  fitted  to  a  set  of  guides.  The  most  common 
fastening  is  shown  in  Fig.  117.  Having  the  forces  W 3  and  W4 
(resultant  forces  from  WJ  acting  on  the  pin  and  allowing  all  the 
reaction  from  the  die  to  fall  at  the  upper  point  of  the  head,  say 
4  in.  above  the  bed,  we  have  a  cantilever  beam  projecting  upward 
from  the  bed  top  and  acted  upon  by  three  forces  tending  to 
break  the  beam  at  some  section  as  ao.  Any  leverages  may  be 


252 


MACHINE  DESIGN 


selected  other  than  4  and  2  but  these  are  given  for  the  sake  of 
argument,  the  actual  values  used  depending  largely  upon  the 
kind  of  die  used  between  the  two  heads.  It  is  evident  that  the 
two  forces  opposing  each  other  (W3,  action  and  reaction)  will 
have  the  same  value.  These  moments  will  cause  stresses  in  any 
section  under  investigation.  Suppose  the  line  ao  to  be  the 
weakest  section  in  the  beam.  The  tendency  to  break  here  is 
resisted  by  two  metal  sections,  each  6  inches  in  width  and  h  inches 
in  height,  or,  by  one  section  2  b  inches  by  h  inches.  The  fiber 
stress  caused  by  the  moments  from  the  two  W3  forces  will  cause 
a  maximum  tension  at  o  which  becomes  less  as  it  approaches  a. 
This  tensional  fiber  stress  at  o  will  be  partially  neutralized  by  a 
downward  force  W4  distributed  more  or  less  uniformly  over  the 
area  2  bh;  the  final  stress  at  o  being  the  algebraic  sum  of  the 
two.  Let  Sp=  pressure  per  square  inch  acting  perpendicularly 


kl 

j 

w*    *t 

I 

r—  £- 

(7 

•+— 

X 

__^  2 

I 

< 

} 

^4 

FIG.  117. 

to  the  bed  top  over  section  2  bh,  Sm  =  tensional  fiber  stress  at  o 
due  to  the  moments  and  St  =  resulting  fiber  stress  at  o. 

Taking  W3  in  two  moments  about  ao  and  W4  in  direct  pressure 
we  have  W4  =  SP  A  and  M  =  Sm  Z,  from- which  we  obtain  Sp  = 

450       ,  ,     _       2552X2X6     15312 

pressure    and   om  = 


due    to    direct 


f-v      7         7  \JL  *-4.Vy  \J\J  \A.JH.\s\j\J  l_/  J.  \_/OO  IAJ.  V/  CVAAVA          ^-'7/2-    OLL2  7^     1*2 

due  to  the  summation  of  the  moments.  Now  if  Sm—Sp  =  St', 
also  if  h  =  5  in.  and  $£  =  1500  we  have  fo^approx.  f  in. 

If  the  fiber  strength  of  tension  and  shear  in  cast  iron  be  taken 
the  same,  then  b'  =  b  approximately. 

In  like  manner  the  reaction  TF3  from  the  die  may  be  taken  at 
the  bottom  instead  of  the  top  of  the  sliding  head,  and  the  turning 
moment  figured  in  this  way  to  see  if  there  is  greater  danger  to 
the  section  than  when  taken  at  the  top. 


TOGGLE-JOINT  DETAILS 


253 


Other  investigations  may  be  made  for  this  fastening.  If  the 
projection  b'  were  fastened  on  with  screws  the  calculations  would 
be  worked  up  in  a  similar  way  to  the  fastening  at  the  base  of  the 
standard. 

Stationary  Head. — As  in  the  sliding  head,  it  is  assumed  tha,t  the 
stationary  head  is  properly  desig- 
ned above  the  bed  top  and  that  the 
fastening  only  is  in  question. 
Fastenings  for  small  machines  will 
not  be  difficult  but  those  for  large 
machines  will  call  for  extreme  care. 
The  simplest  fastening  is  shown 
in  Fig.  118  and  acts  as  a  frictional 
resistance  only.  If  Wt  =  tension  on 
the  bolt  in  pounds,  TF3  =  2552  lb., 

y  =  4  in.,  and  x  =  6  in.,  we  have  by  moments,  disregarding  the 
benefit  obtained  from  the  overlap  of  the  block  around  the 
frame,  2552X4  =  6TF*,  or,  TFf  =  1702  lb.  This  will  hold  the 
block  to  the  bed.  It  is  now  necessary  to  determine  if  the 
block  will  slip  with  this  force  binding  the  frame  between  these 


FIG.  118. 


FIG.  119. 

two  friction  surfaces.  Let  the  coefficient  of  friction  be- 
tween the  block  and  the  frame  also  between  the  washer  and 
the  frame  be  <j>=  say  .3,  then  the  resistance  due  to  friction  is, 
by  formula,  2fiWt  =  F  and  when  applied  to  our  problem  is 
1021.2  lb.  That  is,  with  the  conditions  as  stated,  if  W3  were 


254  MACHINE  DESIGN 

only  40  per  cent  as  large  as  it  now  is  the  block  would  just  slip. 
Since  the  bolt  as  figured  from  moments  proves  to  be  too  small  to 
keep  the  block  from  slipping,  let  us  reverse  the  process  and  find 
how  large  a  bolt  will  be  necessary  to  hold  the  block  against  the 
force  TF3.  By  substituting  as  above  we  have  2  X  .3  X  Wt  =  2552, 
from  which  1^  =  4253.5  Ib.  This  force  is  being  exerted  at  the 
root  of  the  thread  tending  to  elongate  the  bolt.  With  $  =  8000, 
this  will  give  slightly  greater  area  than  .5  sq.  in.  and  will 
require  a  bolt  of  approximately  1  in.  diameter.  It  is  evident 
from  this  that  more  than  one  bolt  should  be  used,  or  that  some 
other  arrangement  be  substituted  for  the  friction  surfaces.  In 
Fig.  119,  A  is  very  similar  to  Fig.  118,  excepting  that  the  lower 

surface  is  notched  to  protect  it 
^  from  slipping.  The  upper  block 

may   slip   slightly,   but   this  will 


cause  a  greater  grip  and  a  conse- 
quent increase  of  frictional  resis- 
tance. A  possible  improvement 
on  this,  if  the  construction  of  the 
machine  would  permit  it,  would 
FIG.  120.  be  to  have  the  bolt  at  an  angle  as 

shown  in  the  dotted  lines. 

Let  this  angle  be,  say  30  degrees  with  the  horizontal,  then  from 
Fig.  120,  A,  .3T  sin  a  =  resistance  due  to  friction,  and  T  cos 
a  =  horizontal  component  of  the  bolt  tension.  Combining  we 
have,  .866r  +  .3x.5T  =  2552,  or, 

77  =  25121b. 

This  will  require  a  lf-in.  bolt. 

Fig.  120,  B,  will  cause  a  tension  on  the  bolt  (disregarding 
friction)  of  T'=2552  tan  (P  +  2).  Let  /?  =  90  degrees  then 
I"  =  2552  Ib.,  requiring  a  if-in.  bolt.  It  is  very  evident  that  if 
friction  were  included  in  this  it  would  reduce  the  bolt  size 
somewhat. 

Let  the  student  investigate  this  with  friction  included. 

C,  Fig.  119  is  probably  not  as  strong  in  the  shape  of  the  tooth 
as  A  and  B,  but  with  a  large  tooth  area  the  unit  shear  becomes 
small  enough  so  that  the  teeth  are  not  endangered.  The  vertical 
faces  on  the  teeth  reduce  the  vertical  thrust  on  the  bolt  to  a 


TOGGLE-JOINT  DETAILS 


255 


minimum  and  permit  the  use  of  a  bolt  just  sufficiently  strong  to 
protect  the  block  from  turning  as  in  Fig.  118. 

D  is  arranged  to  have  pins  to  fasten  into  the  frame  either 
through  the  block,  or  behind  it.  These  pins  keep  the  block 
from  sliding  and  are  calculated  for  shear,  while  the  bolt  is  cal- 
culated to  resist  turning  as  in  Fig.  118. 

Another  way  in  which  these  fastenings  may  fail  is  by  shearing 
the  bolt.     Assume  W3  Fig.  118  entirely  acting  to  shear,  we  have 
2552 

=  '51  sq<  i 


bolt  area< 


taken 


S  =  say   5000 

full  area  of  the  bolt  it  would  be  if  -in.  diameter.  This  shows  a 
requirement  about  equal  to  that  for  tension.  In  any  form  of 
fastening  it  is  well  to  investigate  both  tension  and  shear  and 
take  the  larger  requirement. 

It  should  be  understood  that,  if  the  block  clamps  over  the 
edges  of  the  frame  on  planed  ways,  this  will  assist  the  bolt  in 
holding  the  block  down  and  a  smaller  bolt  may  be  used. 

Let  the  student  investigate  this  as  in  the  case  of  the  sliding 
head. 

135.  Frame  or  Bed.  —  The  calculations  for  the  frame  will  be 
found  somewhat  more  complicated.  Assume  a  simple  type,  say 


FIG.  121. 

of  the  same  general  shape  and  cross-section  as  Fig.  121.  Assume 
also  the  force  W3  acting  at  some  point  along  the  block  face,  say 
at  the  middle  of  the  block,  a  distance  of  2  in.  above  the  top  of 
the  frame.  Any  other  height  may  be  taken  but  in  all  probability 
if  the  dies  should  not  be  parallel  and  they  should  strike  hard  at 
the  top,  this  inequality  would  be  accounted  for  by  a  slight 


256  MACHINE  DESIGN 

springing  of  the  bed.  It  may  be  assumed  that  the  force  W3  will 
act  somewhere  near  the  center  of  the  die  before  it  reaches  such 
a  magnitude  as  to  endanger  the  frame.  This  force  tends  to 
break  the  bed  along  some  line  as  rs,  and  produces  combined  ten- 
sional  and  compressional  stresses  in  the  fibers  of  the  section. 
Considering  the  part  to  the  right  of  the  section  as  free  we  have, 
Fig.  122,  the  fibers  on  the  upper  or  weak  side  subjected  to  two 
tensional  stresses,  the  sum  of  which  should  not  exceed  the  safe 
fiber  stress  of  the  metal,  i.e.,  S1+S2  =  St't  and  the  fibers  on  the 
lower  side,  subjected  to  a  tensional  and  a  compressional  stress, 
the  algebraic  sum  of  which  should  not  exceed  the  safe  com- 
pressional fiber  stress  of  the  metal,  i.e.,  S1  +  (  —  S2)  =SC  where 

$!=  uniform  tensional  stress 

$2  =  stress  due  to  bending 

St  and  Sc  =  combined  stresses. 

To  obtain  Sj^  and  S2  on  the  tension  side  use  the  formulas  W3  = 
S^A  and  M  =  S?Z  and  obtain 

W 

—  p  =  S1  where  A  =  area  of  section  in  square  inches  and 

—  -  —  —  --  -  =  S  where  Z  =  modulus  of  section.     It  will  be  seen 


2 


that   the   moment  arm  is  the  perpendicular  distance  between 
the  force  and  the  center  of  the  section.     The  value  ~  would  be 

£t 

changed  for  any  other  than  a  uniform  section.     See  Art.  144. 

Having  selected  the  section  of  the  bed  as  Fig.  121,  we  find  the 
modulus  to  be 

Z.ty-yvx2  SeeArt.4. 

b  h 

It  will  be  necessary  here  to  select  some  values  for  6,  b',  h  and  h' 
and  make  a  trial  solution.  Take  b=2  in.;  b'  =  1^  in.;  h  =  Q  in. 
and  h'=4:  in.  With  these  values  we  find  A  =  12  sq.  in.  and 

2552 
Sl  =  -^r-  =212.  7  Ib.  per  square  inch 

LZ    ,t 

also   Z  =  18.8   and 

2552  (3  +  2)  .     , 

^2  =  --  TOO  —  -  =  679  Ib.   per  square  inch. 

io.o 

£,  =  ^+£2  =  679+212.7  =  891.7  Ib.    per  square  inch. 


TOGGLE-JOINT  DETAILS 


257 


Since  the  usual  value  of  St  for  cast  iron  is  1500  to  2000,  this 
shape  and  size  of  section  would  be  stronger  than  necessary. 

Now,  if  the  figures  of  the  section  be  changed  to  read  6  =  2  in.; 
&'  =  l^in.;  h  =  5  in.;  and  h'  =4  in.  the  value  becomes 


2552  2552 

«i  =  —  o~~  =  319   Ib.    per   square  inch,  and  S2  =  -- 


=  1126  Ib.  per  square  inch. 

=  Si  =  319  +  1126  =  1445  Ib.  per  square  inch. 


FIG.  122. 

This  seems  to  agree  very  well  with  the  safe  value  of  cast  iron  in 
tension,  and  may  be  used.  Since  this  is  a  symmetrical  section  and 
since  cast  iron  is  much  weaker  in  tension  than  in  compression, 
the  latter  will  not  need  to  be  investigated  and  the  above  figures 
can  be  accepted  for  the  sizes  of  the  bed.  With  a  section  that  is 
unsymmetrical  it  would  be  necessary  to  investigate  both  sides 
of  the  section.  See  Art.  144. 

Having  found  the  shape  of  the  simple  section  it  is  possible  to 
modify  it  to  a  certain  degree  without  affecting  the  calculations 
seriously.  To  illustrate,  the  portion  abed,  Fig.  123,  may  be 
lopped  off  and  added  to  the  inner  side  at  a'b'c'd'  without  affecting 
the  modulus.  Metal  may  be  moved  paralled  to  the  axis  of  the 
section  so  long  as  the  section  is  not  distorted  to  such  an  extent 
that  it  will  break  by  twisting.  Any  change  of  metal,  however, 
toward  or  from  the  axis  of  the  section,  changes  the  modulus  and 
hence  the  resisting  power  of  the  section. 

Fillets  may  be  added  at  the  interior  corners  giving  a  shape 
similar  to  most  frame  tops. 

For  the  bottom,  a  slight  deflection  or  slope  of  the  web,  as  shown 
by  the  dotted  lines,  gives  a  result  very  similar  to  a  plain  cast  iron 


258 


MACHINE  DESIGN 


engine  or  lathe  bed.  Other  minor  changes  such  as  slight  curves 
instead  of  straight  sides  might  be  made  without  any  loss  of 
rigidity.  In  any  case  where  the  shape  of  the  simple  section  is 
found  and  the  designer  wishes  to  increase  the  thickness  of  any 
part  he  may  do  so  and  the  result  is  merely 
to  increase  the  factor  of  safety. 

Suppose  some  other  than  a  uniform 
section  is  desired,  the  same  process  would 
be  employed  in  finding  the  stresses  as 
given  above.  The  modulus,  Z,  however, 
would  be  obtained  as  shown  in  Art.  144. 

If  under  very  heavy  loads  it  is  advisable 
to  specify  one  or  more  steel  I  beams  or 
channels  from  Cambria,  this  may  be  done  by 

making  a  trial  selection  of  a  section  and  substituting  the  value 
of  Z  and  A  in  the  formulas  as  before.  If  this  value  $,  +S2  =  St 
=  8000  to  16,000,  the  exact  value  depending  upon  the  rigidity  of 
the  beam,  the  condition  is  fulfilled  as  in  the  case  of  the  cast  frame. 

136. — The  final  determination  in  this  design  is  to  obtain  the 
length  of  the  frame  to  prevent  overturning  when  the  load  is 


FIG.  123. 


FIG.  124. 


applied.    Let  W  5  Fig.  124  =  the  weight  of  the  frame,  then  from  the 
force  diagram  we  have  the  following  moments  about  the  end  at  b 


but  W'J,3  +  WJ,t  =  W"l3  and  Wx  =  W5l5. 

The  length  may  then  be  obtained  by  adjusting  the  values  of  x 
and  15  such  that  the  equation  will  be  satisfied. 


TOGGLE-JOINT  DETAILS  259 

To  obtain  the  length,  however,  in  a  more  direct  way  the  fol- 
lowing can  be  used: 

If  x  =  l-l3  andZ5=Z2-^2  then  W(l-ls)=WJ,2+2. 

Knowing  the  cross-section  of  the  bed  in  square  inches,  the 
weight  of  1  in.  in  length  would  be  .26 A;  the  total  weight  of  the  bed 
being  .26Z2A  approximately.1  Then  LJ>  =  W(l  -13)  ^.13A. 

Let  I3  =  l2—a  where  a  is  the  offset  as  shown,  then  Z,2  =  TF 
(I  —l2  +  a)  -T-.13A,  from  which  we  obtain  the  formula 


W  W  W^2 

,=  -  3.85-^ ±\   7.7  (l  +  a)  -"-  +  14.82 


weight  of  the  frame  W5  would  be  greater  than  here  shown  be- 
cause of  the  metal  in  the  ends  of  the  frame  and  the  attached  mechanisms, 
all  of  which  would  be  effective.  The  error,  whatever  it  may  be,  is  toward 
that  of  safety. 


260 


MACHINE  DESIGN 


First  Alternate,  Design  No.  1 


w 


FIG.  125,  A 
THE  TOGGLE  JOINT  PRESS 

137.  Assignment. — 

W=  .    .    .    .;  Z=.    .      .    .;  l'=  ....;  6  (min.)  =  .    .    .    . 

In  this  design  the  lever  is  placed  within  the  bed  rather  than 
above  it.  It  will  be  noticed  that  the  end  of  the  bed  is  slotted  to 
allow  for  a  movement  of  the  lever  arm  between  the  points  x  and 
x' '.  The  weakening  of  the  bed  due  to  this  slot  need  not  be  con- 
sidered a  serious  matter.  With  a  long  and  shallow  bed,  however, 
the  movement  of  the  arm  will  be  small  and  will  give  a  very  slight 
movement  to  the  sliding  block.  For  our  purpose  this  machine 
may  be  designed  merely  to  exert  a  pressure  between  the  two 
sliding  blocks,  in  which  case  a  very  slight  movement  is  all  that  is 
necessary  and  the  form  shown  will  be  satisfactory. 


FIG.  125,  B. 

In  case  the  movement  of  the  sliding  block  is  desired  greater 
than  that  allowed  here,  the  lever  may  be  arranged  as  shown  in 
Fig.  125,  B. 


ALTERNATE  DESIGNS  261 

Second  Alternate,  Design  No.  1 


FIG.  126. 
VERTICAL  HAND-POWER  PRESS 

138.  Assignment.  - 

W  =  .  .    .    .    ;  1=    .    .    .    .    ;  I'  =  ....;  6   (mm.)  =    .    .    . 

This  design  follows  the  principles  laid  down  in  No.  1,  with  two 
exceptions.  First,  the  length  I'  here  becomes  so  small  that  a 
separate  crank  cannot  be  used  and  a  bent  shaft  or  an  eccentric  is 
substituted.  In  the  eccentric  the  length  I'  is  the  distance  between 
the  center  of  the  shaft  and  the  center  of  the  eccentric.  Second, 
the  thrust  of  the  sliding  block  is  received  through  a  screw 
directly  against  the  base  of  the  frame.  A  hollow  rectangular 
section  is  suggested  as  the  best  shape  of  the  frame.  Investigate 
also  for  the  screw  and  nut  to  resist  the  thrust. 


262 


MACHINE  DESIGN 
Third  Alternate,  Design  No.  1 


FIG.  127. 
THE  VERTICAL  FOOT-POWER  PRESS 

139.  Assignment. — 

W  =  (100  or  less) Ib. 

I  =  (60  to  72) in. 

V  =  (3  to  6) in. 

6  (min.)  =  degrees. 

This  machine  can  be  used  for  all  kinds  of  light  press  work 
where  but  a  small  movement  of  the  ram  is  needed.  Where  this 
movement  is  desired  as  great  as  possible,  increase  I'  and  decrease 
Z,  also  reduce  the  length  of  the  toggle  members. 

The  ram  may  be  made  rectangular  in  section  and  the  forming 
dies  need  not  be  developed.  The  frame  is  hollow  and  the  lever 
I  is  fastened  on  the  plane  of  the  toggle. 


ALTERNATE  DESIGNS 


263 


Fourth  Alternate,  Design  No.  1 


FIG.  128. 
SMALL  HAND-POWER  PUNCH 

Fig.  128  shows  a  small  bench  tool,  used  for  punching  sheet 
iron  and  other  thin  metals.  Because  of  its  simplicity  only  two 
parts  of  the  assignment  will  be  given.  All  other  necessary 
assumptions  may  be  made  by  the  designer  and  a  complete  set 
of  calculations  and  drawings  made.  The  diagram  to  the  right 
shows  the  mechanism. 

140.  Assignment. — 


W  =  (at  end  of  lever  I,  50  to  100) Ib. 

T  =  (length  of  throat) in. 


264 


MACHINE  DESIGN 
Fifth  Alternate,  Design  No.  1 


FIG.   129. 


HAND-POWER  PUNCH  AND  SHEAR 

The  hand-power  punch  and  shear  is  strictly  a  bench  tool  for 
operating  on  light  work.  The  force  at  the  end  of  the  lever  arm  I 
should  not  be  greater  than  100  lb.;  V  is  the  eccentricity  of  the 
cam,  a  is  the  distance  from  the  pivot  point  of  the  shear  arm  to 
the  point  where  the  cam  force  is  applied,  and  b  is  the  distance 
from  the  pivot  point  to  the  point  of  greatest  shearing  resistance. 

141.  Assignment.  —  (See  Design  No.  2  for  methods.) 
Kind  of  material  to  be  cut  ........................ 

Length  of  cut  or  diameter  of  punch  ................   in. 

Thickness  of  plate  to  be  cut  (up  to  |)  ..............   in. 

Depth  of  throat  ............................   in. 


CHAPTER  XIV 
DESIGN  OF  BELT-DRIVEN  PUNCH  OR  SHEAR 

142.  General  Statement. — A  belt  driven  punch  or  shear  is  the 
machine  selected  to  represent  the  second  general  design.  In- 
cluded within  this  one  machine  are  problems  covering  the  design 
of  frame,  levers,  gears,  fly-wheel,  pulleys,  bearings,  shafts, 
sliding  head,  punch,  die,  clutch,  stripper  and  cam.  The  fact 
that  this  machine  finds  such  general  use  in  manufacturing  plants 
and  that  it  embodies  such  a  variety  of  designs  makes  it  an  ideal 
subject  for  analysis.  Fig.  130  shows  a  motor  driven  shear  of 


FIG.  130. 

late  .design.  It  is  not  expected  that  the  required  design  will  be 
for  a  motor  drive,  but  that  the  distance  between  the  bearings"  be 
shortened  and  pulleys  used  instead.  In  giving  out  the  design 
the  following  requirements  will  be  made:  first,  the  work  to  be 
accomplished,  i.e.,  diameter  and  depth  of  hole  to  be  punched  or 
the  cross-section  of  the  piece  to  be  sheared;  second,  the  maximum 
distance  from  the  edge  of  the  plate  to  the  center  of  the  cutter, 
i.e.,  the  depth  of  the  throat  of  the  machine;  third,  the  average 
cutting  velocity  of  the  punch  or  knife  in  inches  per  second,  ^or 
the  r.p.m.  of  the  cam  shaft. 

265 


266 


MACHINE  DESIGN 


In  the  analysis  of  the  methods  employed  in  working  up  such  a 
design,  the  frame  sections  will  be  carried  out  somewhat  in  detail 
because  of  the  advanced  character  of  the  work;  the  rest  of  the 
machine  will  be  dealt  with  more  briefly.  In  making  the  assign- 
ments, the  members  of  the  class  should  be  given  values  that 
differ  materially  from  those  worked  out  here.  The  five  sample 
plates  at  the  end  of  the  design  show  a  complete  set  of  drawings 
of  such  a  machine. 

143.  Requirements  of  the  Design. — A  machine  to  punch  a  1-in. 
hole  through  f-in.  mild  steel  plate,  the  center  of  the  hole  to  be 
not  greater  than  7  in.  from  the  edge  of  the  plate.     The  velocity 
of  the  punch  during  cutting  may  be  taken  in  this  case  as  approxi- 
mately 1  in.  per  second. 

144.  Frame. — The    material   used   in   the   frame    of   such    a 
machine  is  either  close  grained  cast  iron  or  steel  casting.     The 
general  shape  is  about  as  shown  in  Fig.  130  and  the  sections  of  the 
frame,  Fig.  131,  are  either  hollow  cast  iron  as  shown  in  B  and  C 
or  web-shaped  steel  as  shown  in  A.     Of  the  three  sections,  B  and 
C  are  the  most  common.     Fig.  137  represents  the  outline  of  the 


nyf- 

I  <w 

*• 

~jj-* 

a 

l 

\^//^/ 

|p2fc 

y- 

"C 

•—  -y  —  » 

WT"~ 

•=-1-" 

-* 

'^y^SM 

A      b    i 

»   t 

»  g 

FIG.  131. 

assembly  drawing  as  finally  worked  out  about  x  x,  the  center  line 
of  the  frame.  To  plan  the  general  shape  of  the  frame  about  the 
punch,  begin  by  laying  off  the  throat  depth,  G,  say  8  in.,  along 
the  line  x  x.  Find  H  of  the  same  figure  by  assuming  some  shape 
of  frame  section  and  calculating  the  sizes  for  the  various  parts 
of  the  section  as  described  in  this  article.  Find  also  other  safe 
sections  at  various  angles  to  the  horizontal  and  trace  the  outer 
curve  <of  the  frame  through  the  outer  points  of  these  sections, 


PUNCH  AND  SHEAR  DETAILS 


267 


after  which  plan  the  speed  mechanism  and  locate  the  shafts. 
It  is  necessary  many  times  to  modify  the  first  layout  a  great 
deal  but  this  must  be  expected  and  should  not  cause 
discouragement. 

To  work  out  the  sizes  of  the  horizontal  section  along  x  x,  select 
the  shape,  say  B,  Fig.  131,  from  the  standard  forms  and  apply 
the  method  used  in  Art.  135,  taking  G  as  the  depth  of  the  throat 
and  y  as  the  distance  from  the  edge  of  the  casting  to  the  center 
of  gravity  of  the  section. 

In  applying  the  formula  # 2  =  — — exercise  care  in  obtain- 

A  (t  or  c) 

ing  a  satisfactory  value  for  Z  in  the  unsysmmetrical  section.  To 
get  Z  it  will  be  necessary  to  determine  the  moment  of  inertia  I  of 
the  section  and  then  find  Z  by  the  following : 


for  tension  Zt  = 


for  compression  Zc  =  — 


Make  a  trial  selection  of  some  sizes  for  the  section  and  find  the 
gravity  axis  by  cutting  out  a  pasteboard  section  and  balancing 
it  upon  knife  edges;  or  a  better  way  is  by  the  following:  assume 


any  line  of  reference  as  ab,  take  the  algebraic  sum  of  the  moments  of 
each  rectangular  section  about  this  line  of  reference  and  divide  by 
the  total  area;  this  will  give  the  distance  x  between  the  line  of  reference 
and  the  gravity  axis  gg  of  the  section,  ab  may  be  taken  at  any, 
position  in  the  section  but  the  work  will  be  much  simplified  if  it  \ 
is  taken  at  the  edge  or  at  the  center  of  the  section.  When  gg 
is  determined  find  I  by  the  following:  to  the  sum  of  the  products 


268  MACHINE  DESIGN 

of  each  area  by  the  square  of  the  distance  from  its  center  of  gravity 
line  to  the  gravity  axis  of  the  section  add  the  moment  of  inertia  of 
each  section  about  its  own  gravity  axis.  It  will  be  remembered  that 
the  moment  of  inertia  of  any  rectangle  about  its  own  gravity  axis 
is  /  =  6/t3-j-12  where  h  =  the  total  height  of  the  section. 
Assume  the  section  with  s&es  as  shown  in  Fig.  132  then 

70X11.5-2X10X10 

70+20  +  54"       =4'2m- 


7=70X(7.3) 

14X(5)s     3X(18)3     10X(2)3 
~YT        ~12~        ~12~ 

Zt=—      '—  =  1054  for  tension 
y.o 

=  679  for  compression. 

-.i 

W  is  the  pressure  on  the  punch  in  pounds.  If  the  ultimate 
shearing  stress  of  mild  steel  be  taken  at  55,000  Ib.  per  square 
inch,  W  would  be  129,591  Ib.  Considering  the  .trial  section  only 
on  the  tension  side,  since  this  is  usually  the  weak  side  of  the 
section,  we  have  Sl+S2  =  St  =  900  +  2189  =  3089  Ib.  per  square 
inch.  This  fiber  stress  would  be  large  for  cast  iron  in  tension, 
hence  another  section  must  be  selected. 

Take  for  a  second  trial  the  section  Fig.  133,  we  have,  if  worked 
as  above 

x  =  2.97  in. 

7  =  2i;049.44 

„_  |  1680  for  tension 

\  1317  for  compression. 

£i+£2  =  £*  =  2154  Ib.  per  square  inch. 

In  like  manner  we  should  work  out  the  compression  side  by  Sl  — 
S2  =  SC.  The  algebraic  sum  of  the  two  gives  571-2013  =  -  1442 
Ib.  per  square  inch.  The  sign  may  be  considered  either  positive 
or  negative  since  it  merely  indicates  the  direction  in  which  the 
force  acts  and  does  not  affect  the  magnitude  of  the  force.  See 
also  Art.  135. 


PUNCH  AND  SHEAR  DETAILS 


269 


Any  other  section  of  the  frame  can  be  determined  by  finding 
$2  in  the  manner  shown  above  and  combining  with  it  the  value  of 
S1  =  W  cos  a  -T-  area.  The  value  of  Sl  is  a  maximum  when  a  is 
zero  and  becomes  zero  when  a  is  90°.  It  will  be  seen,  Fig.  134, 


FIG.  134. 

that  at  the  section  A,  S,  =  W -=- area  A',  at  B,  S1  =  Fb-^&Te&  B, 
but  Fb  =  W  cos  a  hence  SV  =  W  cos  a  -f- area  B;  at  C,  Sl  =  W  cos 
a  -4-  area  C  and  so  on  until  St  becomes  zero  at  section  E.  At  this 
point  the  frame  should  be  examined  for  both  bending  and  shear- 
ing and  the  larger  requirement  taken.  In  all  probability  section 


270 


MACHINE  DESIGN 


E  will  be  made  larger  than  the  calculated  size  to  accommodate 
the  finishing  around  the  head.  It  will  be  satisfactory  in  this 
design  if  we  obtain  sections  at  a  =  0,  45  and  90  degrees. 

To  find  any  section,  say  a  =  45  degrees,  determine  the  height  of 
the  gap,  k,  and  draw  the  outline  of  the  gap.     The  value  k  is  con- 
trolled by  the  space  taken  up  by  the  dies,  metal  to  be  punched, 
and  clearance.     It  cannot  be  determined  exactly,   but  a  good 
estimate  may  be  made.     Assuming  some  section  of  the  frame 
as  Fig.  135  and  solving  for  the  fiber  stress  as  before,  we  find 
x  =  2.67  in. 
1  -15,655. 
Zt  =  1382. 

Sl+S2  =  St  =  1872  Ib.  per  square  in. 

NOTE. — In  finding  M  in  M  =  SZ  the  lever  arm  varies,  depending 
upon  the  cosine  of  the  angle  a. 

Investigate  also  for  Sc. 


o 

28"- 


FIG.  135. 


For  the  vertical  section  take  Fig.  136,  in  which  case  we  have 
re  =  3. 26  in. 
7  =  4411.79 
Zt  =  655. 

S2  =  791  Ib.  per  square  inch,  which  shows  that  the  section 
could  be  materially  reduced  in  size  if  it  were  desired.  The 
reduction  could  very  properly  be  made  according  to  the  dotted 
lines.  If  it  were  considered  necessary,  this  section  should  also  be 
investigated  for  compression. 

To  investigate  for  shearing  on  the  vertical  section  we  have, 


PUNCH  AND  SHEAR  DETAILS 


271 


allowing  the  shear  to  be  absorbed  by  the  entire  section  of  136 
sq.  in., 

„  .     , 

Ib.  per  square  inch. 


129591 


OC 
lob 


FIG.  136. 

145.  Having  determined  several  important  sections  in  the 
frame,  the  outline  of  the  G  part  of  the  frame  can  be  drawn.  This 
outline  will  of  course  be  modified  somewhat  for  the  shaft,  head 
and  leg. 

It  will  be  noticed  that  a  somewhat  higher  fiber  stress  has  been 
allowed  in  the  material  for  this  frame  than  in  the  material  used  in 
the  frame  of  the  first  design.  This  is  about  as  would  be  expected. 
Any  casting  planned  to  fill  a  very  important  place  in  the  design 
of  any  machine  would  be  made  of  the  best  close  grained  gray 
iron.  It  is  advisable  to  keep  the  size  of  this  frame  as  small  as 
possible  consistent  with  strength  and,  since  the  best  of  cast  iron 
would  have  an  ultimate  strength  of  25,000  to  30,000  Ib.  per 
square  inch,  it  would  be  considered  safe  to  allow  a  fiber  stress  of 
2000  to  2500  Ib.  per  square  inch,  which  corresponds  to  a  factor  of 
safety  of  12. 

The  shape  of  the  section  may  be  varied  to  suit  the  conditions, 
from  a  large  and  thin  section  as  here  treated,  to  a  small  compact 
and  possibly  solid  section.  The  latter  condition  prevails  in  some 
machines  where  the  gap  is  long  and  the  main  section  would  be 
necessarily  crowded  into  the  smallest  space. 

Steel  cast  frames  are  very  common,  especially  on  the  larger 
machines.  When  made  of  steel  the  frame  section  may  be  made 
much  smaller.  St  =  12,000  to  15,000  Ib.  per  square  inch. 


272 


MACHINE  DESIGN 


Tension  bars  are  provided  for  machines  with  long  gaps.     These 
bars  are  very  necessary  when  doing  heavy  duty. 


FIG.  137. 


146.  The  Maximum  Punching  or  Shearing  Force  is  used  in 
calculating  the  frame  sections.  The  ultimate  shearing  stress  of 
the  metal  multiplied  by  the  area  to  be  cut  gives  the  maximum 
load  on  the  punch  or  flat  shear.  If  the  maximum  load  on  a 
bevel  shear  is  desired,  multiply  the  maximum  load  on  a  flat 
shear  by  the  following: 


4°  Bevel 
8°  Bevel 


THICKNESS   OF  THE   METAL 

fill*       U       If       li  If  If  1|  '       2 

.42     .48     .54     .61      .67     .73     .79  .85  .92  .98 

.23     .3       .37     .44     .51      .58     .65  .73  .81  .88      .95 


Look  up  articles  on  the  Shearing  of  Metals  in  the  American 
Engineer  and  Railway  Journal,  Vol.  LXVII,  page  142. 

In  any  machine  of  this  kind  it  is  safe  to  allow  15  to  20  per  cent 
for  the  friction  of  the  parts  while  performing  the  heaviest  duty. 


PUNCH  AND  SHEAR  DETAILS 


273 


The  total  pressure  to  be  accounted  for  at  the  driving  end  in  this 
machine  will  then  be  129,591^.85  =  152,460  Ib.  If  the  eccen- 
tricity of  the  cam  be  taken  the  same  as  the  thickness  of  the 
thickest  metal  to  be  punched,  =f  in.,  the  twisting  moment  on 
the  main  shaft  will  be  approximately  JX  152,460  =  114,345 
inch  pounds. 

147.  Working  Depth  of  the  Cut. — The  actual  cutting  depth 
(depth  of  penetration)  of  a  punch  or  flat  shear  may  be  used  in 
determining  the  foot  pounds  of  work  done  at  the  tool,  and  is  a 
certain  percentage  of  the  total  thickness  of  the  metal.  Generally  the 
tool  in  its  movement  passes  entirely  through  the  metal,  but  the 
work  of  cutting  is  finished  when  the  tool  arrives  at  the  depth  of 
penetration.  This  percentage  varies  somewhat  with  the  kind 
of  the  metal,  but  for  mild  steel  it  has  been  found  by  experiment 
(Am.  Mach.,  Oct.  12,  1905)  to  be 


Thickness  of  metal,  in  inches  1 

Depth  of  penetration  in  per  cent   of 


I       I       T56    i    T3«       i    A     A     A 


plate  thickness 


25  31  34  37  44  47  50  56  62  67  75  87 


Thus  the  work  of  cutting  is  finished  when  the  punch  (or  flat 
shear)  has  reached  a  depth  of  .25X1  =  .25  in.  in  a  1-in.  plate, 
.185  in.  in  a  J-in.  plate,  .125  in.  in  a  J-in.  plate,  and  so  on. 

148.  Diameter,    Width   and   R.P.M.    of   the   Pulleys.— Table 
LVIII  gives  values  agreeing  fairly  well  with  current  practice  for 

TABLE  LVIII 


Machine  will  punch 

Diameter  of  pulley 

R.p.m. 

iin.Xj  in. 

10 

200  to  250 

§  in.  Xi  in. 

12 

200  to  250 

|  in.  X  fin. 

16 

175  to  200 

1  in.  X  1  in. 

18 

150  to  175 

2  in.  X  1  in. 

30 

150  to  175 

274  MACHINE  DESIGN 

the  diameter  and  revolutions  per  minute  of  the  pulleys.  To 
determine  the  width  of  the  pulley  face,  or  the  width  of  the  belt, 
no  definite  rule  can  be  stated.  Practice  varies  between  a  2-in. 
belt  on  a  |-in.  X  i-in.  machine,  and  a  6-in.  or  7-  in.  belt  on  a  2-in. 
X  1-in.  machine.  Calculations  for  belt  sizes  on  such  machines  do 
not  give  very  satisfactory  results  because  of  the  small  percentage 
of  each  revolution  that  the  machine  is  actually  working.  It  is  a 
good  experience,  however,  if  each  man  would  apply  a  few  trial 
conditions  and  note  the  results.  First  find  the  effective  pull  P 
on  the  belt,  by  the  horse-power  formula  or  by  moments  from  the 
cam  shaft,  assuming  the  punch  or  shear  to  be  cutting  full  value 
all  the  time,  and  then  take  the  percentage  of  this  which  is  repre- 
sented by  the  proportion  of  the 'total  time  that  the  cutter  is 
actually  working.  Figure  the  belt  from  this  result  as  in  Art. 
117.  In  all  probability,  catalog  sizes  will  finally  be  taken. 

149.  Fly-wheel.  Weight. — The  weight  of  the  fly-wheel  may 
be  obtained  by  either  one  of  two  methods;  first,  by  assuming 
the  wheel,  when  running  at  full  speed,  to  have  stored  up  energy 
enough  to  do  a  certain  definite  work;  second,  that  the  wheel  shall 
have  only  a  certain  allowable  fluctuation  from  full  load  to  no  load. 
From  the  first  method,  a  fly-wheel  for  a  machine  of  this  kind  may 
be  designed  to  fulfill  a  number  of  conditions,  from  a  wheel  such 
that  its  kinetic  energy  will  just  equal  the  energy  absorbed  by  the 
machine  during  punching  (in  which  case  if  we  disregard  the  belt's 
action,  the  velocity  of  the  wheel  would  become  zero  after  each 
hole  punched),  to  a  wheel  of  such  a  size  that  the  residual  energy 
will  be  sufficient  to  keep  the  speed  fairly  constant.  Current 
practice  approaches  the  former  and  in  this  consideration  will  be 
adopted. 

Having  given  the  force  to  be  accounted  for  at  the  driving  end 
as  152,460  lb.,  assume  that  this  force  acts  through,  say  a  maxi- 
mum of  one-half  the  total  depth  of  the  plate,  f  in.  or  ^  ft.,  then 
the  energy  exerted  would  be  4764  foot  pounds.  Apply  the 
formula  Wv2-^-2g  to  the  mean  rim  diameter,  where  W  =  weight 
of  wheel  in  pounds  assumed  centered  at  the  center  of  the  rim 
and  v  =  velocity  at  any  point  in  this  circumference  in  feet  per 
second.  Assuming  36  in.  as  this  diameter  with  150  r.p.m.  (see 
Art.  150)  we  have 


PUNCH  AND  SHEAR  DETAILS 


275 


Wv2 


=  4764;  TF  =  5531b. 


The  depth  of  penetration  in  this  application  is  not  used  according 
to  the  table.  This  should  not  be  confusing  since  it  is  merely  for 
illustration. 

Find  the  weight  of  the  fly-wheel  also  from  some  acceptable 
formula  based  upon  the  fluctuation  of  speed,  using  for  the  allow- 
able fluctuation  20  to  25  per  cent,  and  check  with  the  above. 

Arm.— The  fly-wheel  arm  may  be  calculated  as  follows: 
estimate  the  time  required  in  punching  one  hole,  then  find  the 
distance  through  which  a  point  on  the  center  line  of  the  rim  will 
move  during  this  time;  this  will  be  the  value  V  in  PF  =  4764. 
Since  the  shaft  is  running  15  r.p.m.,  each  revolution  will  take 


FIG.   138. 

four  seconds.  Assuming  the  velocity  of  the  punch  during 
action  to  be  the  same  as  that  of  the  cam  center  we  have  3.1416 
X 1 .5  -T-  4  =  1 . 1781  in.  per  second.  The  time  occupied  in  punching 
'is  |  -7-1.1781  =  .318  second.  The  velocity  of  the  rim  of  the  wheel 
is  1413.7  ft.  per  minute  =  23. 56  ft.  per  second,  from  which  we 
find  that  the  rim  will  travel  7.5  ft.  before  stopping. 
Applying  P7  =  4764, 

P  =  6351b. 

The  value  of  P  may  be  found  in  another  way.  First,  with  the 
radius  of  the  large  gear  =  22. 5  in.,  find  the  force  p  between  the 
gears,  Fig.  138.  From  the  moments  around  the  cam  shaft,  this 
is 


276  MACHINE  DESIGN 

129591X3 


and  by  moments  around  the  driving  shaft 

5082X9 
^18~= 

Having  found  P,  the  tractive  force  due  to  the  stored  up 
energy  of  the  wheel  rim,  obtain  the  large  dimension  of  the  arm 

at  the  center  of  the  shaft  by  the  formula  ~^r  =  .05&3$.     If  N, 
the  number  of  arms,  =  6  and  $  =  1500, 


635X18     _3  in 
6X.  05X1500 

A  low  fiber  stress  is  used  because  of  unknown  stresses  that  are 
apt  to  be  in  the  casting.  Straight  arms  are  preferred  to  curved 
arms  and  they  should  have  well-rounded  fillets  next  to  the  hub 
and  rim.  The  section  of  the  arm  near  the  hub  and  that  at  the 
rim  are  always  similar.  The  dimensions  at  the  rim  should  be 
taken  not  less  than  two-thirds  of  the  corresponding  dimensions 
at  the  hub.  The  ordinary  arm  has  the  thickness  at  the  center 
of  the  section  about  one-half  of  the  length  of  the  section.  The 
radius  of  the  side  of  the  section  is  about  three-fourths  of  the 
longest  dimension  of  the  section.  The  value  b  as  given  in  the 
formula  is  sometimes  taken  at  the  center  of  the  shaft  and  fre- 
quently at  the  edge  of  the  hub.  This,  it  will  be  seen,  makes 
very  little  difference  in  the  average  pulley. 

150.  Driving  Shaft. — If  the  bearings  are  close  to  the  pulley 
and  gear  the  bending  will  not  be  excessive  and  the  shaft  may  be 
figured  with  a  low  fiber  stress  merely  to  resist  twisting.  Taking 
$  =  6000,  the  diameter  of  the  shaft  will  be  2.2,  say  2\  in. 

On  machines  where  the  pull  of  the  belt  and  the  side  thrust 
from  the  gears  are  fairly  great,  also  when  the  bearings  are  far 
apart,  it  is  ne'cessary  to  design  the  shaft  for  combined  twisting  and 
bending.  In  such  a  case  find  the  side  thrust  due  to  each,  the  belt 
and  the  gears,  and  calculate  the  shaft  from  the  bending  moment 
as  a  beam  fixed  at  the  ends  and  loaded  at  two  points.  See 
Art.  4. 


PUNCH  AND  SHEAR  DETAILS 


277 


In  locating  the  shaft  DD,  it  is  first  necessary  to*  have  the 
approximate  position  of  the  main  shaft  and  the  diameters  of  the 
gears.  Knowing  the  angular  velocities  of  the  two  shafts  the 
diameter  of  the  small  gear  may  be  assumed  and  the  distance 
between  the  shaft  centers  obtained.  In  this  machine  if  the 
cutting  speed  of  the  punch  is  1  in.  per  second,  the  center  of  the 
cam  will  travel  approximately  60  -=-4.71  =  13  revolutions  per 
minute.  Calling  this  15  and  the  revolutions  per  minute  of  the 
pulley  shaft  150  the  ratio  of  the  gears  is  10.  With  4J  in.  as  the 
diameter  of  the  pinion,  the  shafts  will  be  24f  in.  between  centers. 

151.  Gears. — Design  according  to  Art.   92  for  machine  cut 
teeth.     The  pinion  should  be  shrouded.     The  diameter  of  4J 
in.,  as  used  in  Art.     150,  is  merely  for  illustration.     This  value 
would  be  rather  small  for  the  construction  of  a  perfect  tooth. 
The  arms  of  the  large  gear  are  similar  to  those  in  the  fly-wheel 
excepting  that  the  driving  force  will  be  absorbed  by  not  more 
than  one-half  the  number  of  arms  in  the  gear. 

152.  Main  Shaft. — The  main  shaft  or  "cam  shaft7'  as  it  is  some- 
times called  would  be  made  of  hammered  steel.     Figs.  139  and 
140  show  two  common  forms.     Bl}  B2  and  B3  are  journals,  and 
C  is  the  cam  which  operates  the  punch.     The  greater  part  of  the 
thrust  from  the  punch  is  absorbed  at  the  journal  B2,  B3  being 
added  for  the  double  purpose  of  reducing  the  strain  of  the  shaft 
and  for  an  outside  connection  for  adjustments. 


rz£  H 

*  —  ii  —  • 

^/ 

/*\ 

X 

C 

~, 

FIG.  139. 

In  designing  the  shaft  the  part  a  may  be  figured  to  resist  the 
twisting  moment  due  to  the  thrust  on  the  gear  x,  allowing  a  fiber 
stress  of,  say,  6000  Ib.  per  square  inch  for  shear.  It  will  be 
noticed,  however,  that  the  thrust  on  the  gear  produces  a  bending 

moment  on  the  shaft,  the  lever  arm  being  —     This  bending 


278  MACHINE  DESIGN 

moment  may  be  of  such  magnitude  as  to  make  it  necessary  to 
use  the  combined  formula.  It  would  be  well  to  obtain  the 
diameter  from  both  formulas  and  check  them. 

The  length  of  the  journal  may  be  taken  from  2  diameters  to 
2.5  diameters  of  the  shaft.  The  length  of  b  will  be  quite  variable 
and  will  be  governed  by  the  frame  of  the  machine.  The  diam- 
eter of  b  will  depend  upon  the  judgment  of  the  designer.  In 
some  shafts  it  is  made  equal  to  the  diameter  of  the  left  journal 
while  in  others  it  is  enlarged  to  the  size  of  the  main  journal.  A 
high  speed  machine  would  require  a  larger  and  stiffer  shaft  than 
a  slow  speed  machine,  because  of  the  heavy  shocks  to  which  the 
shaft  is  subjected,  hence  the  diameter  of  b  would  be  as  large  as 
possible. 

Take  the  size  of  the  main  journal  such  that  the  pressure  per 
square  inch  of  projected  area  will  not  exceed  3000  Ib.  assuming 
the  entire  thrust  from  the  punch  to  be  taken  up  by  it  and  that 
of  the  cam  not  to  exceed  8000  Ib.  Lower  values  than  these  are 
desirable,  especially  on  the  cam,  where  5000  Ib.  per  square 
inch  of  projected  area  is  a  good  value.  It  will  be  seen  from  the 


FIG.  140. 

above  that  the  projected  area  being  constant,  a  bearing  may  be 
changed  in  shape  decidedly  and  yet  give  good  service.  As  an 
illustration  B2  may  be  long  and  slender  as  in  Fig.  139,  or  short  and 
thick  as  in  Fig.  140,  so  long  as  the  shaft  at  this  point  is  stiff 
enough  to  resist  bending  and  shear.  Conditions  within  the 
machine  itself  usually  determine  the  shape  of  bearing  and  cam. 
When  the  sizes  are  approximately  determined,  they  should  be 
constructed  graphically  to  scale,  usually  having  the  two  surfaces 
continuous  along  one  line  as  at  x. 

The  cam  varies  from  3  to  6  in.  in  length,  and  from  6  to  12. in. 
in  diameter.  The  diameter  of  the  bearing  in  such  a  case  is 
governed  somewhat  by  the  eccentricity  of  the  cam. 

The  cross-sectional  area  of  the  bearing  J52  along  its  outer  face 


PUNCH  AND  SHEAR  DETAILS  279 

next  the  cam  must  be  sufficient  to  resist  the  effect  of  shear;  it 
must  also  resist  the  bending  moment  produced  by  the  thrust 


multiplied  by  the  half  length  of  the  cam  ( -  I  and  the  torque 

w 

produced  by  the  thrust  multiplied  by  the  eccentricity  of  the 
cam.  This  should  be  worked  by  the  combined  formula,  remem- 
bering that  B3}  where  used,  would  reduce  this  bending  moment 
somewhat. 

In  machines  where  the  distance  between  Bx  and  B2  is  great 
there  is  a  bending  of  the  shaft  between  the  bearings.  This  is 
especially  true  where  B3  is  omitted  as  in  some  horizontal  machines. 
Such  a  condition  is  equivalent  to  a  beam  in  flexure  with  the 
reactions  at  B1  and  C  and  the  applied  load  at  B2.  The  effect, 
however,  is  not  the  same  in  the  calculations  as  a  simple  beam 
because  of  the  support  given  to  it  by  the  boxes. 

It  is  safe  to  assume  that  the  bearings  are  sufficiently  loose  to 
allow  some  bending,  but  not  loose  enough  to  consider  it  as  a 
simple  beam.  Probably  a  safe  assumption  would  be  50  per  cent 
of  the  maximum  load  applied  at  the  cam  center  and  resisted  at 
the  bearing  centers  as  supports. 

The  frame  should  be  fitted  with  a  phosphor-bronze  bushing  J 
in.  to  f  in.  in  thickness  surrounding  the  journal  B2.  This 
bushing  is  made  a  forced  fit  with  the  frame. 

The  sizes  of  B3  would  vary  between  2  in.  and  4  in.  for  both 
diameter  and  length. 

Application. — Calculating  the  shaft  for  twist  at  its  smallest 
diameter,  at  the  gear,  gives  d  =  4.59,  say  4.5  in. 

The  cam  diameter,  assuming  a  length  of  4  in.  and  a  pressure 

129591 

per  square  inch  of  5000  Ib.  is  Knnn — 7  =  6.5  in 

oUUUX  4 

B2  will  then  be  5  in.  diameter  and,  if  we  allow  2500  Ib.  per 

129591 

square  inch  projected  area,  will  have  a  length  of        n — ^  =  10.4 

^oUUX  o 

in.,  say  11  in. 

B3  may  be  taken  2|  in.  long  by  3  in.  diameter. 

153.  Sliding  Head. — Of  the  different  types  in  use,  two  of  the 
very  common  ones  are  shown  in  Figs.  141  and  142,  the  former 
being  used  in  the  smaller  machines.  The  chief  objection  to  the 


280 


MACHINE  DESIGN 


bronze  block  is  its  liability  to  wear  unevenly  thus  causing  lost 
^motion  and  an  irregular  movement  of  the  block  while  punching. 
In  the  latter  form,  the  entire  thrust  is  carried  on  a  hardened  steel 
block  set  into  the  cast  iron  sliding  head  and  the  wear,  if  any,  is 
practically  uniform.  The  size  of  the  bearing  surface  in  the  steel 


block  may  be  obtained  from  the  crushing  strength  of  the  steel 
casting.  If  this  value  be  taken  at  90,000  Ib.  per  square  inch 
with  a  factor  of  safety  of  6,  the  projected  area  of  this  bearing 
will  be  129,591^15,000  =  8.6  sq.  in.,  from  which,  if  the  length 

of  the  cam  be  4  in.,  the  breadth  of  the 
bearing  will  be  2.15  in.  say  2J  in.  The 
breadth  of  the  sliding  head  face  will  be 
seen  to  depend  upon  the  construction 
of  the  vibrating  arm.  Make  the  vibrat- 
ing arm  a  steel  casting  and  allow  from 
J  in.  to  1J  in.  at  x,  and  a  small  clearance 
at  y.  This  part  of  the  work  must  be 
done  graphically.  The  values  a  and  b 
will  depend  respectively,  upon  the  width 
of  the  frame  and  the  diameter  of  the 


FIG.  142. 


bolts  used. 

* 

154.  Clutches  and  Transmission  Device. — In  operating  any 
machine  having  an  intermittent  motion  a  clutch  is  commonly 
used  to  serve  as  a  connector  between  the  power  supply  and  the 
work.  The  application  of  the  clutch  to  the  simple  punching  or 
shearing  machine  is  shown  in  Fig.  143.  It  is  usually  applied 
directly  to  the  hub  of  the  large  gear  and  is  operated  through  a 
system  of  levers  and  cranks  by  either  hand  or  foot.  When  the 
punch  is  not  operating,  the  large  gear,  which  is  designed  with  a 
long  hub  to  act  as  a  bearing,  runs  loose,  the  shaft  remaining 


PUNCH  AND  SHEAR  DETAILS 


281 


stationary.  The  clutch  sleeve  slides  on  the  shaft  over  a  splined 
key  and  when  the  punch  is  to  be  operated  this  sleeve  is  thrown 
to  engage  with  the  corresponding  part  on  the  gear  hub.  When 
the  hole  is  punched  a  counterweight  brings  the  sleeve  back  to 
its  former  position  and  the  movement  of  the  punch  ceases. 

Clutches  are  formed  each  having  two,  three,  or  four  j  aws.  These 
jaws  may  be  formed  as  a  part  of  the  wheel  hub  as  shown  at  A 
and  B,  cast  from  steel  and  bolted  to  the  flat  face  of  the  wheel 
hub  as  shown  at  C,  or  cast  from  steel  and  fitted  to  the  interior  of 
the  wheel  hub  as  shown  at  G.  In  heavy  work  C  and  G  are 
preferable. 

That  part  of  the  clutch  subjected  to  the  greatest  wear  is  the 
front  face  of  the  jaw.  This  is  sometimes  fitted  with  a  plate  of 
high  carbon  steel  which  can  be  replaced  when  necessary  with  a 
new  one.  The  rear  face  of  the  jaw  is  usually  perpendicular  to 
the  front  face  of  the  wheel  but  is  sometimes  cut  to  an  angle  of 
30  to  45  degrees.  There  should  be  sufficient  clearance  between 
the  jaws  on  the  sleeve  and  the  wheel  to  enable  them  to  be  easily 
thrown  together  while  in  motion.  This  should  be  from  £  in. 
to  J  in. 

The  clutch  sleeve  may  be  shaped  as  shown  in  either  D  or  F. 
The  following  sizes,  table  LIX  will  meet  average  requirements. 

TABLE  LIX 


Shaft  = 

2  in. 

3  in. 

4  in. 

5  in. 

6  in. 

a 

1 

1 

u 

1* 

If 

b 

i 

1 

H 

li 

U 

c                                          /                                                (a  +  b) 

d 

4 

5* 

7 

9 

10i 

e 

5 

7 

9 

11 

12 

f 

3 

4 

5* 

7 

8 

g 

t 

1 

H 

If 

li 

282 


MACHINE  DESIGN 


There  are  two  general  methods  of  designing  the  transmission 
device;  the  first  and  simpler  one  E  having  the  clutch  between  the 
gear  and  the  frame,  and  the  second  H,  having  the  gear  between 


PUNCH  AND  SHEAR  DETAILS 


283 


the  clutch  and  the  frame.  The  latter  method  necessitates  a 
hollow  shaft  in  order  to  obtain  a  rigid  connection  between  the 
sleeve  and  the  clutch  and  is  not  much  used  on  small  machines. 

155.  Punch,  Die,  and  Holders. — In  all  punch  and  die  work  the 
die  is  made  a  little  larger  than  the  punch  for  clearance.     The 


r> 


— 

'ri 


E 


rrnJ  SLIDINC 

.    HEA'JD.  vn-n 

T! 
m      V, 

PUNCH    < 

->^SMiLAT»  .  ]  j 

1  1 
|  I 

!    ^ 

1 

ij 

. 

1             \ 

1  !"  "!i: 

TTJ 

LIT 

D 

Uuj      jib 

'  i 

1      o 

D 

i 

a 


FIG.  144. 


action  of  the  punch  on  the  material  is  shown  in  A,  Fig.  144,  the 
hole  tapering  from  the  size  of  the  punch  on  one  side  to  the  size 
of  the  die  on  the  other.  This  taper  is  slight  and  is  considered  of 
no  consequence  in  rough  work,  but  in  finished  work  it  is  a  difficulty 
that  can  easily  be  remedied  by  reaming  the  hole  afterward.  For 
reference  see  "  Dies,  Their  Construction  and  Use.  "  Woodworth. 


284  MACHINE  DESIGN 

There  are  various  methods  of  fastening  the  punch  to  the 
sliding  head;  B  shows  the  bottom  of  the  sliding  head  fitted  with 
the  square  ended  socket  and  punch.  A  screw  ended  socket  is 
sometimes  used  as  at  E.  C  shows  the  bottom  of  the  head  flanged 
and  drilled  for  the  attachment  of  either  punches  or  shears.  In 
single  machines  it  is  desirable  that  both  punching  and  shearing 
be  done.  Where  such  is  the  case  this  is  a  good  form.  Side 
adjustment  of  the  punch  may  easily  be  made  if  the  head  be 
slotted  as  at  D  and  fitted  with  a  tee  block  as  E.  Dies  are  made 
from  high  carbon  steel  and  are  held  in  a  holder;  the  holder  in 
turn  is  bolted  to  the  horizontal  face  of  the  frame.  A  certain 
amount  of  adjustment  is  necessary  in  locating  the  die,  conse- 
quently the  holder  is  made  in  two  parts. 

Other  Types  of  Shearing  and  Punching  Machines 

The  smallest  sizes  of  punching  and  shearing  machines  are  oper- 
ated by  hand  power  or  foot  power,  medium  sized  machines  are 
operated  almost  exclusively  by  belt  and  the  largest  machines  are 
operated  by  belt,  steam,  water  or  electricity  as  shown  in  Figs. 
145,  146,  147,  and  148  respectively.  These  designs  show 
present  practice  and  are  added  to  enable  the  designer  to  become 
more  familiar  with  the  form  of  the  parts  and  the  make-up  of  the 
machines  in  general. 

It  will  be  noticed  that  in  the  larger  machines  the  frame  is  of 
such  a  size  as  to  project  below  the  floor,  the  weight  being  carried 
on  legs  or  lugs  cast  on  the  side  of  the  frame.  It  will  also  be 
noticed  that  arrangements  are  made  at  the  top  of  the  frame  for 
the  attachment  of  a  crane  to  assist  in  handling  the  material. 

Most  single  machines  have  the  lower  end  of  the  ram  so  con- 
structed that  either  punches  or  shear  blades  may  be  attached. 
This  requires  some  little  time  in  changing  and  adjusting  the  tools. 
Double  machines  avoid  the  necessity  of  such  changes. 

Machines  such  as  are  here  represented  require  more  work  than 
should  be  expected  of  one  assignment.  They  may,  however,  be 
assigned  to  two  men.  This  is  especially  true  of  the  double 
machines,  in  which  case  the  frames  may  be  worked  up  independ- 
ently, and  the  driving  mechanism,  jointly.  Electric  motor  sizes 
and  capacities  may  be  obtained  from  any  standard  catalog  of 
electric  machinery. 


LARGE  PUNCHING  AND  SHEARING  MACHINES          285 


FIG.  145. 


FIG.  146. 


286 


MACHINE  DESIGN 


FIG.  147 


FIG.  148. 


TYPICAL  DRAWINGS 


287 


I 


288 


MACHINE  DESIGN 


TYPICAL  DRAWINGS 


289 


11 

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^   <  «  ^       ^ 

Ibl]  I 

y Q   ^    t 

&  I&-  ^ 
z  If  §- 
en  Cs  ^ 


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O 

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290 


MACHINE  DESIGN 


\ 


. 


PLATE  G  7 


SINGLE  POWER  PUNCH 
DETAILS 


Pvrdue  University  Lafayette  Ind 


I 


. . .    . 


ALTERNATE  DESIGNS 


First  Alternate,  Design  No.  2 


291 


B  — 


FIG.  149 

THE  BEVEL  SHEAR 
(Niles-Bement-Pond   Catalog) 

156.  Assignment.— 

Kind  of  material  to  be  sheared 

Width  of  plate  to  be  sheared  (6  to  12) in. 

Thickness  of  plate  to  be  sheared  (J  to  1) in. 

Depth  of  throat  (6  to  18) in. 

Strokes  of  the  ram  per  minute  (15  to  20) 

The  frame  sections  may  be  calculated,  if  desired,  to  a  regular 
outline  as  shown  in  the  dotted  lines,  after  which  modifications  in 
this  outline  may  be  made  by  approximation.  A  better  way, 
however,  would  be  to  sketch  the  approximate  longitudinal  frame 
section  as  above  and  figure  for  each  of  the  several  irregular 
sections,  as  A,  B  and  C. 


292 


MACHINE  DESIGN 


Second  Alternate,  Design  No.  2 

K,\'Y\'<v^- 

JB 


FIG.  150. 

HORIZONTAL  POWER  PUNCH 
(Niles  Tool  Works  Co.  1900  Catalog) 


157.  Assignment— 

Kind  of  material  to  be  punched 

Size  of  largest  hole  punched 

Thick*  ?ss  of  the  plate 

Distance  of  center  of  hole  from  edge  of  plate 
Number  of  holes  punched  per  minute 


Horizontal  punching  machines  may  be  designed  in  the  same 
gei.jral  way  as  the  one  described  in  the  notes.  It  will  be  found 
that  the  frame  sections  may  be  calculated  in  the  same  way 
although  the  frame  not  being  so  regular  will  require  a  little  more 
care  in  selecting  the  shapes  and  sizes  of  the  various  parts  of  the 
sections. 

Machines  of  this  type  usually  have  a  more  shallow  throat  than 
the  vertical  type. 

The  line  of  the  punch  center  may  be  raised  from  the  center  of 
the  ram  to  the  upper  edge  and  is  found  convenient  when  punching 
near  a  shoulder. 


ALTERNATE  DESIGNS 
Third  Alternate,  Design  No.  2 


293 


FIG.  151. 
THE  BULLDOZER 

158.  Assignment— 

Length  of  stroke  (6  to  18) in. 

Maximum  pressure  (5000  to  30,000) Ib. 

Number  of  strokes  per  minute  (10  to  15) 

The  Bulldozer,  one  of  the  most  powerful  of  the  horizontal 
presses,  is  used  in  forming  or  squeezing  metals  to  shape  between 
large  dies  in  such  processes  as  upsetting  and  bolt  heading.  It  is 
also  occasionally  used  in  punching  and  straightening.  The  dies 
are  very  heavy,  and  sometimes  the  stroke  is  made  long  encagh 
to  permit  a  number  of  dies  being  inserted  at  one  time,  so  as  to 
allow  several  operations  on  the  specimen  without  reheating. 

Assume  a  typical  work  card,  having  the  ordinates  represent 
total  pressures  in  pounds  and  the  abscissas  represent  per  cents  of 
stroke.  Let  the  work  to  be  performed  be  such  that  the  dies  first 
strike  the  specimen  at  25  per  cent  of  the  stroke,  also  let  it  require 
the  following  total  pressures  to  complete  the  work : 

Per  cent  of  stroke 25  30  35  40  50  60  70  80  90  95  100 

Per  cent  of  max.  pressure  00  60  70  75  80  82  78  75  75  80  100 


294 


MACHINE  DESIGN 
Fourth  Alternate,  Design  No.  2 


FIG.  152. 


ATLAS  POWER  PRESS 
(Atlas  Machine  Co.  Catalog) 

The  press  shown  in  Fig.  152  is  designed  to  take  the  place  of  the 
ordinary  foot-press  in  doing  light  blanking,  perforating,  riveting, 
forming  and  closing.  The  clutch  is  of  the  standard  Johnson  type. 
A  ball-and-socket  joint  between  the  shaft  and  the  gate  gives  the 
latter  a  vertical  adjustment  of  about  1J  in.  The  machine  is 
furnished  with  a  combination  pulley  and  balance  wheel.  The 
mechanism  of  the  machine  is  shown  to  the  right.  The  following 
approximate  sizes  may  be  used  for  checking: 

From  bed  to  gate  in  lowest  position 6  to  7  in. 

Stroke li  in. 

Distance  between  uprights 3§  to  6  in. 

Bed  surface 7  X 10  to  8  X 12  in. 

Weight  of  wheel 50  to  100    Ib. 

159.  Assignment. — 

P  (crank  5  degrees  from  vertical,  1000  to  2000) Ib. 

T  (depth  of  gap)  (4  to  6) in. 

Revolutions  per  minute  (200  to  300) 


ALTERNATE  DESIGNS  295 

Fifth  Alternate,  Design  No.  2 


FIG.  153.  —  THE  LENNOX  ROTARY  SHEAR. 

(Joseph  T.  Ryerson  Catalog) 
(Bethlehem  Foundry  and  Machine  Co.  Catalog) 

160.  Assignment.  — 

Kind  of  material  to  be  sheared  ........................... 

Depth  of  throat  (6  to  36)  ................................  in. 

Thickness  of  plate  (|  to  f  )  ...............................  in. 

Diameter  of  cutters  (6  to  12)  .............................  in. 

Rim  velocity  of  cutters  (600  to  1000  ft.  per  hour)  ...........  • 

NOTATION 

Shaft  L  is  adjustable  at  lower  end  by  screw  H  around  R  as  a 
pivot.  See  Fig.  155. 

Shaft  N  is  adjustable  in  line  parallel  to  center  of  shaft  by  nut 
T.  J  and  0  are  gears  of  same  diameter.  The  main  pulley  of  the 
machine  runs  from  150  to  250  revolutions  per  minute.  The 
cutters  at  D  may  be  set  apart  a  distance  as  great  as  one-fourth 
the  thickness  of  the  plate;  the  exact  amount  can  best  be  deter- 
mined by  the  experience  of  the  operator.  The  exact  force  W  at 
the  cutters  tending  to  rupture  the  frame  is  rather  an  indeterminate 
quantity  but  a  safe  value  may  be  found  by  the  following  formula: 


where  t  =  thickness  of  the  metal  to  be  cut  ;  r  =  radius  of  the  cutters  ; 
/=the  ultimate  shearing  strength  of  the  metal;  and  A  =  area  of 
the  metal  being  cut  at  any  time,  assuming  the  cutters  to  be  in 
contact  at  the  center  line. 


296 


MACHINE  DESIGN 


FIG.  154. 
LENNOX  ROTARY  BEVEL  SHEAR 


FIG.  155. 
SECTION  OF  SHEAR 


ALTERNATE  DESIGNS 
Sixth  Alternate,  Design  No.  2 


297 


FIG.  156. — SHEET  METAL  FLANGER  AND  Disc  CUTTER. 
(Niagara  Machine  and  Tool  Works  Catalog) 


The  section  shows  the  machine  with  flanging  rolls. 
These  may  be  changed  to  cutter  rolls  as  shown  at  the 
left.  Other  small  rollers  hold  the  metal  to  the  plate 
while  being  operated  upon.  Sizes  of  flanges  obtained 
in  soft  sheet  steel  as  follows:  10  to  16  guage,  |-  to  1 
in.;  16  to  20  gage,  f  to  f  in.;  22  to  24  gage,  %  in. 
Machine  will  cut  up  to  No.  8  gage  and  flange  to  No. 
10  guage. 

161.  Assignment. — 

P  =  (1000  to  2000) Ib. 

T  (depth  of  throat  on  machine,  12  to  20)..  .  .  in. 
G  (depth  of  throat  on  circle  arm,  30  to  40) . .  in. 
Speed  of  the  tool  (15  to  20)  .  .  .rev.  per  minute. 


298 


MACHINE  DESIGN 
Seventh  Alternate,  Design  No.  2 


BARTLETT  &  CO., 


FIG.  158. — BOILER  HEAD  FLANGING  MACHINE  DETAILS. 
(Niles-Bement-Pond  Catalog) 

162.  Assignment. — This  problem  consists  essentially  of  the 
development  of  the  mechanism  and  the  design  of  the  parts  shown 
in  the  detailed  figures,  i.e.,  the  flanging  mechanism.  For  applica- 
tion of  these  parts  to  the  machine,  see  catalog.  Develop  the 
mechanism  so  that  the  rollers  will  revolve  about  each  other  with  a 
uniform  clearance  in  all  positions.  Assume  a  maximum  thrust 
at  the  roller  of  (10,000  to  100,000)  pounds,  and  design  the  parts 
so  they  will  be  sufficiently  rigid  to  protect  from  flexure  and 
breaking,  and  so  the  pull  on  the  hand  wheel  will  be  within  the 
capacity  of  one  man,  say  150  Ib. 


CHAPTER  XV 
DESIGN  NO.  3 


FIG.  159. 


THE  AIR  HOIST.     (Whiting  Foundry  Equipment  Co.) 

163.     Assignment. — Capacity  in  free  load Ib. 

Weight    of   parts   and  friction per   cent 

Air  pressure  (80  to  100) Ib.  per  sq.  in. 

Lift  (2  to  4) ft. 

The  following  data  may  be  used  for  checking,  air  at  80  Ib.  per  square  inch. 


Diam.  of  hoist 

Size  of  pipe 

Air  consumed  per  4-ft.  lift 

3  in. 

?  in. 

1  .  17  cu.  ft. 

Tin. 

fin. 

6.63  cu.  ft. 

10  in. 

1  in. 

13.50  cu.  ft. 

16  in. 

1  in. 

34.49  cu.  ft. 

299 


300 


MACHINE  DESIGN 

First  Alternate,  Design  No.  3 

THE  ALLEN  RIVETERS 
(Joseph  T.  Ryerson  and  Son  Catalog) 


FIG.  160. 
Lattice  column  type. 


FIG.  161. 
Jaw  type. 


In  this  type  of  machine  the  piston  rods  connect  levers  of  differ- 
ent lengths,  thus  forming  a  toggle  joint.  It  very  properly 
embodies  features  of  both  designs,  No.  1  and  No.  2.  It  may  be 
assigned  as  an  advanced  substitute  for  No.  1  or  as  an  extra. 

The  following  maximum  pressures  necessary  to  set  rivets  may 
be  expected  in  average  practice. 


Diameter  of  rivets  in  inches 
Pressure  in  tons  of  2000  Ib. 


4       I       i       I       1     H     H       U 
25     30     40     50     65     80     100  150 


ALTERNATE  DESIGNS 


301 


FIG.  162. 


164.  Assignment. — 


P  =  (8000  to  50,000) 

p  =  (Air  pressure  80  to  100) . 

T  =  (4to  12)  L.  C.  type 

T  =  (IQ  to  66)  J.  type 

1= 

e  (mm.)  =  . . 


Ib. 

Ib.  per  sq.  in. 

in. 

in. 

in. 

in. 


302 


MACHINE  DESIGN 


Second  Alternate,  Design  No.  3 


SECOND  POSITION  -TOGGLE  ACTION 
COMPLETED  AND  LEVER  ACTION  BEGUN 


THIRD  POSITION  -  LEVER  ACTION 
AND  STROKE  COMPLETED 


FIG.  163. 


FIG.  164. — MECHANISM. 


THE  HANNA  RIVETER 


FIG.  165. — DEVELOPMENT  OF  THE  MECHANISM. 


ALTERNATE  DESIGNS 


303 


FIG.  166. 


165.  Assignment.  — 
P  =  (50,000  to  200,000) 


lb. 
in. 

Maximum  movement  of  the  die  (0  +  N)  ...................  in. 

Air  pressure  ...................................  lb.  per  sq.  in. 

Assign  Plunger  Travel,  0  +  N 
Approximate  other  dimensions  to  table 


A 
9 

B 
12 

C 
1 

D 

7i 

D' 

9 

E 

18 

F 
26 

M 

8f 

N 
3* 

0 

i 

P 
12 

First  calculate  and  obtain  the  sizes  for  the  frame,  then  give  lengths  and 
locate  levers  EA,  BF,  CG  and  DH  such  that  the  first  half  of  the  piston 
movement  will  cause  a  constantly  decreasing  velocity  of  the  die,  and  the 
last  half  of  the  piston  momement  will  cause  a  uniform  velocity. of  the  die. 
As  an  illustration  of  the  above:  in  one  machine  the  piston  movement  was, 
12  in.,  the  total  movement  of  the  die  was  4  in.,  the  first  5  in.  of  piston 
travel  gave  a  constantly  decreasing  velocity  of  the  die  through  3^  in.  of  the 
die  movement,  leaving  the  last  7  in.  of  piston  movement  to  produce  a  uni- 
form velocity  of  the  die  through  the  last  half  inch  of  die  movement. 


304 


MACHINE  DESIGN 
Third  Alternate,  Design  No.  3 


FIG.  167. 

THE  ALLIGATOR  RIVETER 
(Jos.  T.  Ryerson  and  Son.     Catalog) 

166.  Assignment. — (See  Notation  on  First  Alt.  No.  3.) 

P  =  (25  to  65) tons. 

p  =  (air  pressure,  80  to  100) Ib.  per  sq.  in. 

IT  =(9  to  14) in. 

6  (min.)  = degrees. 

Maximum  movement  of  the  dies  (2  to  4) in. 

Assume  the  length  of  the  arm  of  the  scissors  such  that  the  force 
to  be  transmitted  through  the  toggle  will  not  be  so  great  as  to 
require  too  large  a  cylinder.  Also  observe  that  a  long  arm 
requires  a  long  toggle  link  and  hence  a  long  piston  movement. 
This  style  of  machine  is  used  largely  in  structural  and  car  shops. 
It  may  be  made  vertical  or  horizontal  type.  The  dies  are  adjust- 
able. The  height  of  the  gap  varies  from  6  to  14  in. 


ALTERNATE  DESIGNS 
Fourth  Alternate,  Design  No.  3 


305 


FIG.  168. 

MUDRING  RIVETER 

167.  Assignment. — 

W  =  (25,000  to  100,000) Ib. 

p]  =  (air  pressure,  80  to  100)   Ib.  per  sq.  in. 

T  =B  =  (8  to  16) ' in. 

A  =  (5  to  8 in. 

C  (min.)  when  6  = degrees. 

Total  die  movement  (3  to  5) in. 

This  design  may  be  modified  by  having  the  cylinder  enclosed 
within  the  base  if  desired.  In  such  an  arrangement  the  piston  rod 
becomes  a  compression  member.  Design  also  for  air  pipes  and 
valves. 


306 


MACHINE  DESIGN 


Fifth  Alternate,  Design  No.  3 


FIG.  169. 
LEVER  RIVETER 


168.  Assignment. — 

W  =  (20,000  to  60,000) Ib. 

p   =   (air  pressure,  80  to  100) Ib.  per  sq.  in. 

T  =  (throat,  8  to  12) in. 

A  =  (20  to  36) I in. 

C  (min.)  when  6  = degrees. 

Total  die  movement  (2  to  3) in. 

When  the  arms  are  in  their  inner  positions  the  cylinder  must 
not  touch  them.     Design  also  for  air  pipes  and  valves. 


ALTERNATE  DESIGNS 
Sixth  Alternate,  Design  No.  3 


307 


FIG.  170. 
25  Ton  Portable. 


FIG.  171. 
50  Ton  Portable. 


HYDRAULIC  RIVETING  MACHINE 

(Niles-Bement-Pond  Catalog) 

169.  Assignment. — 

P=  (15-50) tons. 

p  =  (800-1500) Ibs.  per  sq.  in. 

T  =  (6-15) in. 

Size  of  rivet  (see  First  Alt.  Des.  No.  3) in. 

In  this  design  the  cylinder,  frame,  supports  and  valves  are 
important  in  the  order  named.  The  piping  is  a  feature  that  can 
be  modified  to  suit  almost  any  condition  of  frame.  Such 
machines  are  used  on  structural  and  bridge  work. 


308 


MACHINE  DESIGN 


Seventh  Alternate,  Design  No.  3 

TRIPLE  PRESSURE  HYDRAULIC  RIVETING  MACHINE 
(Niles-Bement-Pond  Company) 

This  machine  is  built  with 
three  capacities:  50,  100 
and  150  tons,  for  driving  f-, 
1^-and  1^-in.  rivets.  The 
gap  T  is  made  in  two  lengths 
12  and  17  ft.  The  cylinder 
is  designed  for  three  pres- 
sures of  water,  the  highest 
being  1500  to  2000  Ib.  per 
square  inch.  By  means  of 
the  three  pressures  pro- 
vided as  per  section  (see  also 
catalog)  the  distributing 
valve  is  not  needed. 

On  frame  B  is  mounted 
cylinder  A  with  main 
plunger  E.  To  the  main 
plunger  can  be  attached,  by 
means  of  the  interrupted 
thread  and  nut  J,  the  small 
plunger  7.  When  so  ar- 
ranged plungers  E  and  / 
move  together  and  pressure 
on  the  dies  is  equivalent  to 
the  pressure  p  on  the  differ- 
ence between  the  two  areas. 
Small  plunger  can  also  be 
attached  to  main  plunger 
so  that  intermediate  sleeve 
FIG.  172. 

H  is  locked  to  main  plunger 
and  moves  with  it.  When 
so  arranged  the  pressure 
on  the  dies  is  controlled  by 
the  difference  between  area 
of  main-plunger  and  inter- 
mediate sleeve  H.  Cover  F 
over  die  slide  D  contains  the 
push  back  piston  G  bearing 
directly  upon  main  plunger. 

170.  Assignment. — 
P  =  (50-150)  tons.        p 


FIG.  173. 
(1500-2000)  Ibs.  persq.  in. 


T  =  (12-17)  ft. 


CHAPTER  XVI 

STUDIES  IN  THE  KINEMATICS  OF  MACHINES. 

The  following  problems  in  kinematics  are  given  to  supplement 
the  work  in  both  mechanism  and  design.  One  or  more  of  these 
problems  may  be  assigned  between  designs  1  and  2,  also  between 
2  and  3  and  will  serve  as  a  relaxation  from  the  tedium  of  the  longer 
problems  of  design.  In  their  solution  they  contemplate  pure 
mechanism  (line  motion)  only  and  will  not  deal  in  any  way  with 
the  strength  or  proportion  of  parts.  The  problems  are  arranged 
in  a  graduated  series:  first,  those  distinctly  outlined  and 
requiring  little  or  no  originality;  second,  those  open  to  original 
ideas,  but  having  one  solution  suggested,  out  of  a  number  that 
might  be  made;  third,  those  open  to  a  number  of  solutions  but 
requiring  complete  originality  and  invention. 

A  series  of  illustrative  problems  in  the  study  of  the  mechanical 
movements  of  machines  was  given  in  the  Am.  Mach.,  one 
problem  each  month,  beginning  December  1,  1904.  In  order  that 
originality  be  developed,  it  is  suggested  that  these  problems  be 
read  in  connection  with  the  assignment  given. 

The  kinematic  problems  relating  to  valve  gears  and  link 
motions  are  classified  at  the  last  of  the  list  and  may  be  given 
between  designs  2  and  3  or  after  design  3,  so  as  to  be  taken  in 
parallel  with  or  after  the  subject  of  Engines  and  Boilers. 


309 


310 


MACHINE  DESIGN 
Kinematic  Sheet  No.  1 


Approximcrfe 
Epicycloid 


Approximcrte 
Epicycloid 


Exact 
Epicycloid, 


Exact 
Epicycloid 


FIG.  174. 

171.  Assignment. — Given,  a  problem  by  which  the  diameters 
of  the  gears  may  be  obtained.  It  is  required  to  construct  the 
tooth  outlines  for  each  by  means  of  the  following  systems: 

Exact  epicycloidal 
Exact  involute 
Approximate  epicycloidal 
Approximate  involute. 


KINEMATIC  PROBLEMS 
Kinematic  Sheet  No.  2 


311 


0  '.    j4re  of  firclc. 
b.   Arc   of  cam. 

FIG.  175. 
PLANER  CAM 

172.  Assignment.  — 

A  and  C  are  loose  pulleys,  B  is  a  tight  pulley. 

D  is  fastened  to  frame  of  planer. 

I  moves  back  and  forth,  oscillating  link  F  about  G. 

E  is  rigidly  connected  to  F  by  set  screw. 

Levers  0  and  P  are  pivoted  at  m  and  n  on  Z). 

Q  and  R  are  rollers  fastened  to  the  shifting  levers. 

Pulley  diameters  (10-24) 

S  = 

Width  of  fast  pulley  (3-10) 

Width  of  loose  pulleys,  each,  (2-8) 

Construct  curve  of  cam  so  that  the  shifter  will  be  constantly 
accelerated  during  first  half  of  its  motion  and  constantly  retarded 
during  latter  half. 

During  the  first  half  of  the  motion  of  E  (or  F),  one  shifter  arm 
moves  outward,  while  the  other  arm  remains  stationary  (in  the 
outward  position).  During  the  second  half  of  the  motion  of  E, 
the  second  shifter  arm  moves  inward,  while  the  first  arm  remains 
stationary  (in  the  outward  position)  .  Place  the  points  Q  and  R 
respectively  directly  above  and  below  the  center  of  rotation  of 
the  cam  E. 


in. 
in. 
in. 
in. 


312 


MACHINE  DESIGN 
Kinematic  Sheet  No.  3 


CAM  OF  HOME  SEWING  MACHINE 
173.  Assignment. — 

Diameter  of  cylinder  (1^-3$) in. 

Depth  of  groove  (f^— i)   in. 

Diameter  of  roller  (t3g-|) in. 

Stroke  of  bar  (1-3) in. 

Length  of  arm  G  (6-10) in. 

Length  of  arm  H  (10-18) in. 

Design  a  cylindrical  cam  similar  to  that  shown  in  the  sketch 
to  engage  a  rocker  arm.  Divide  the  motion  into  24  time  periods. 
The  follower  is  to  move  with  a  constant  acceleration  during  four 
time  periods;  during  the  next  eight  periods  it  is  to  move  uniformly 
with  the  velocity  attained;  during  the  next  four  periods  it  is  to 
come  to  rest  with  a  constant  retardation.  The  return  motion  con- 
sists of  eight  time  periods;  during  the  first  four  periods  it  is  to  be 
constantly  accelerated  and  during  the  remaining  four  periods  it 
is  to  be  constantly  retarded. 

Required  full  projection  of  cam  outline  on  the  cylinder.  This 
will  require  the  development  of  the  cylinder  at  top  and  bottom  of 
groove. 


KINEMATIC  PROBLEMS 
Kinematic  Sheet  No.  4 


313 


FIG.  177. 

SEWING  MACHINE  BOBBIN  WINDER 
174.  Assignment.— 

Number  of  threads  to  be  laid  per  inch  of  spool  length  (30-100) 
Length  of  spool  (l$-2i) 


314 


MACHINE  DESIGN 


Kinematic  Sheet  No.  5 


FIG.  178. 

175.  Assignment.— 

Design  the  constant  diameter  cam,  A,  as  shown,  under  the  fol- 
lowing conditions:  follower  to  move  with  harmonic  motion 
from  extreme  right  to  left;  to  return  one-half  the  distance  by 
uniform  motion;  to  remain  at  rest  for  one-sixth  the  revolution  of 
the  cam,  and  to  return  to  starting  point  by  uniform  motion. 
Total  stroke  of  follower  in  one  direction  =  .  ,  .  .  in. 


KINEMATIC  PROBLEMS 


315 


Kinematic  Sheet  No.  6 


FIG.  179. 
QUICK  RETURN  MECHANISM 

176.  Assignment. — 

Length  of  lever,  A    (18-24) in. 

External  diam.  of  circular  slot    (8—10) in. 

Distance  from  center  of  rotating  shaft,  F,  to  center  of  circular 
slot   (4-10) in. 

Plot  velocity-time  diagram  of  crosshead  at  end  of  arm  A,  which  moves 
along  a  horizontal  line  through  F. 


316 


MACHINE  DESIGN 


Kinematic  Sheet  No.  7 


FIG.  180. 


QUICK  RETURN  MECHANISM 

177.  Assignment.— 

Radius  of  pin  B  from  A  (8-16) in. 

Distance  from  A  to  D   (18-24) in. 

Distance  of  A  above  horizontal  line  through  D in. 

If  B  revolves  with  uniform  rotation  about  A,  plot  the  velocity-time 
diagram  of  block  at  lower  end  of  EG. 


KINEMATIC  PROBLEMS 
Kinematic  Sheet  No.  8 


317 


FIG.  181. 

178.  Assignment. — Lay  out  a  Whitworth  Quick  Return  mo- 
tion, with  the  path  of  the  tool  below  the  center  B  of  the  slotted 
crank  BP,  according  to  the  following  data: 

Length  of  stroke in. 

Length  of  connecting  rod in. 

Length  of  A.  P in. 

R.p.m.  of  crank 

Period  of  advance  to  return  of  tool  =  2:1 

Construct  the  linear  velocity-space  diagram  of  the  tool. 


318 


MACHINE  DESIGN 


Kinematic  Sheet  No.  9 


F 


FIG.  182. 

179.  Assignment. — Assume  center  A  directly  above  center  C; 
also  that  slot  in  which  B  works  is  on  the  arc  of  a  circle,  with 
radius  A  B.  Plot  velocity-time  diagram  for  member  F  if  AB 
rotates  continuously  and  members  are  proportioned  as  follows: 

Length  AB    (6-12) in. 

Length  CD  (18-30) in. 

Length  DE  (16-20) in. 


KINEMATIC  PROBLEMS 


319 


Kinematic  Sheet  No.  10 


FIG.  183. 

180.  Assignment. — Having  given  an  oscillating  arm,  pivoted 
at  point  B,  design  a  cam  to  move  the  end  of  the  arm  over  the 

path  1,  2,  3,  4,  5 13.     The  cam  may  have  a 

uniform  or  varying  motion  while  the  arm  may  move  uniformly 
or  according  to  any  law  of  motion  desired. 


320 


MACHINE  DESIGN 
Kinematic  Sheet  No.  It 


FIG.  184. 


181.  Assignment. — Two  crossheads  are  to  be  driven  in  paths 
A B  and  CD  intersecting  at  right  angles.  The  length  of  the 
stroke,  CD,  is  one-half  that  of  A  B.  Motion  is  to  be  given  to 
both  crossheads  by  a  single  rotating  cam.  Such  guides  and 
connecting  rods  as  are  necessary  may  be  employed.  No  part 
of  the  mechanism  is  to  project  within  the  angle  DAB  at  any  time. 
Motion  away  from  A  is  to  be  according  to  the  following  schedule: 

I  stroke,  uniform  acceleration. 

|  stroke,  uniform  motion. 

i  stroke,  uniform  acceleration. 

Motion  toward  A  to  be  harmonic. 


KINEMATIC  PROBLEMS 
Kinematic  Sheet  No.  12 


321 


FIG.  185. 

182.  Assignment. — Let  vertical  crosshead  be  A,  horizontal 
crosshead  be  B,  the  pin  connection  be  C,  then  C  will  travel 
through  the  stationary  cam  curve  as  shown. 

Length  of  horizontal  connecting  rod in. 

Length  of  vertical  connecting  rod in. 

Travel  of  horizontal  crosshead in. 

Travel  of  vertical  crosshead in. 

Crossheads  to  move  out in.  with  uniform  acceleration; 

out in.  with  uniform  motion;  out in.  with 

uniform  acceleration;  and  to  move  in in.  with  increasing 

harmonic  motion;  in in.  with  uniform  motion  and  in 

in.  with  decreasing  harmonic  motion. 

Develop  both  top  and  bottom  of  groove  in  cone  cam. 


322 


MACHINE  DESIGN 


Kinematic  Sheet  No.  13 


FIG.  186. 

183.  Assignment. — Having  given  the  path  of  a  groove  ABC, 
a  follower  block  is  to  move  from  A  to  B  to  C  to  B  to  A.  Design 
a  mechanism  without  the  use  of  cams,  and  without  allowing  any 
part  of  the  driving  mechanism  to  extend  within  the  angle  ABC. 
Rack  and  pinion,  or  chain  drives  cannot  be  used  directly  to 
produce  the  motion. 


KINEMATIC  PROBLEMS 


323 


Kinematic  Sheet  No.  14 


<ua/atf 

S/idinS  Block. 


FIG.  187. 


184.  Assignment: — Having  given  any  path  ARST  around 
which  a  point  is  to  travel,  design  a  mechanism  to  guide  the  point, 
the  mechanism  to  have  but  one  rotating  shaft  and  one  rotating 
disc  cam,  although  other  machine  elements  may  enter  into  the 
construction.  No  part  of  the  mechanism,  excepting  a  single 
driving  arm,  shall  project  within  the  path  ARST,  or  above  the 
horizontal  line  drawn  through  T. 

The  movement  of  the  point  will  be 

A  to  R  (         )  of  period  of  rotation. 

II  to  S  (         )  of  period  of  rotation. 

S  to  T  (         )  of  period  of  rotation. 

T  to  A  (         )  of  period  of  rotation. 


324  MACHINE  DESIGN 

Original  Kinematic  Problems 

185.  Assignment. — Sketches  A  to  E  show  some  of  the  common 
forms  of  paper  clips  on  the  market.  The  problem  is  to  design 
cams,  connecting  levers  and  properly  shaped  dies  to  produce 
from  a  spool  of  wire  some  one  of  the  forms  indicated.  Sketches 
may  be  taken  as  full  size. 


FIG.  188. 

186.  Assignment. — The  path  of  a  block  consists  of  two  parts, 
A B  and  BC.     BC  is  J  the  length  of  A B  and  perpendicular  to  AB. 

Motion  cycle  to  be  as  follows: 

J-  B  to  A,  uniform  acceleration. 

f  B  to  A,  uniform  motion. 

J  B  to  A,  uniform  acceleration. 

A  to  B,  harmonic  motion. 
J  B  to  C,  uniform  acceleration, 
f  B  to  C,  uniform  motion. 
J  B  to  C,  uniform  acceleration. 

C  to  B,  harmonic  motion. 

The  motion  of  the  block  is  to  be  obtained  from  a  single  disc 
cam,  and  no  part  of  the  mechanism — excepting  a  single  guiding 
arm  to  impart  motion  to  block — shall  extend  outside  the  angle 
ABD}  where  D  is  on  a  continuation  of  CB.  Use  not  more  than 
two  levers  or  bell  cranks  and  no  connecting  links,  and  have 
block  make  complete  cycle  in  one  revolution  of  cam. 

187.  Assignment. — The  path  of   a  block  is  to  be  a  square 
A,  B,  C,  D}  the  block  to  be  driven  by  a  single  cylindrical  cam 
rotating  with  a  vertical  shaft,  i.e.,  shaft  is  perpendicular  to  plane 


KINEMATIC  PROBLEMS  325 

of  path.     No  part  of  the  driving  mechanism  is  to  operate  in  the 
plane  of  the  square.     The  motion  cycle  is  to  be: 

£  A  to  B,  uniform  acceleration. 

f  A  to  B,  uniform  motion. 

\  A  to  B,  uniform  acceleration. 

This  to  be  repeated  for  B  to  C,  C  to  D,  and  D  to  A. 

188.  Assignment. — A  follower  block  has  motion  along  a  path 
ABCD.     AB  and  DC  are  each  perpendicular  to  BC,  on  the  same 
side,  and  at  the  ends  of  the  line  BC.     In  length,  these   path 
parts    bear    the    following  relations:    BC  =  2AB  =  \\DC.     One 
cylindrical  cam  is  to  be  used,  and  no  part  of  the  driving  mechan- 
ism is  to  extend  within  the  figure  ABCD,  at  any  time  during  the 
motion,  the  cycle  of  which  is  to  be: 

i  B  to  C,  constant  acceleration. 

J  B  to  C,  uniform  motion. 

J  B  to  C,  constant  deceleration. 

-J-  C  to  D,  constant  acceleration. 

J  C  to  D,  uniform  motion. 

•J-  C  to  D,  constant  deceleration. 

D  to  C,  same  variations  as  B  to  C. 

C  to  5,  same  variations  as  B  to  C. 

B  to  A,  A  to  B,  harmonic  motion. 

189.  Assignment. — A  follower  block  is  to  move  in  a  groove 
whose  center  line  is  ABC.     BC  is  perpendicular  to  AB,  and  }  as 
long  as  A  B.     The  motion  is  to  be  given  by  a  single  cylindrical 
cam,  which  may,  however,  carry  more  than  one  groove.     Not 
more  than  two  levers  or  bell  cranks  and  not  more  than  two 
connecting  rods  may  be  used.     No  part  of  the  mechanism  is  to 
extend  within  the  angle  ABC,  and  the  cam  must  lie  in  the  angle 
made  by  prolonging  AB  and  CB. 

4  A  to  B,  constant  acceleration. 
•J  A  to  B,  constant  motion. 
J  A  to  B,  constant  deceleration, 
f  B  to  C,  increasing  harmonic. 
J  B  to  C,  constant  motion. 


326  MACHINE  DESIGN 

|  B  to  C,  decreasing  harmonic. 
J  C  to  B,  constant  acceleration, 
i  C  to  5,  constant  motion. 
£  C  to  #,  constant  deceleration. 
J  B  to  A,  increasing  harmonic. 
%  B  to  A,  constant  motion. 
J  B  to  A,  decreasing  harmonic. 

190.  Assignment. — Block  as  follower  to  move  in  groove  whose 
center  line  is  ABC. 

Block  to  be  driven  by  two  disc  cams  on  the  same  shaft. 

No  part  of  mechanism,  excepting  a  single  driving  arm,  to 
extend  inside  the  angle  ABC  or  above  the  line  ABD.  Mechanism 
to  be  sufficiently  substantial  and  positive  for  die  work. 


B 
FIG.  189. 


Motion  to  be  as  follows: 


\  B  to  Ar  uniform  acceleration. 

\'B  to  A,  uniform  velocity. 

\  B  to  A ,  uniform  acceleration  (decreasing). 

A  to  B,  harmonic  motion  (increasing  and  decreasing) . 

B  to  C,  same  as  the  first  three  above  from  B  to  A. 

C  to  B}  harmonic  motion  (increasing  and  decreasing) . 

191.  Assignment. — Block  as  follower  working  in  slot  whose 
center  line  is  ABC.  To  be  driven  by  two  cylindrical  cams  with 
axes  at  right  angles  to  each  other.  No  part  of  the  mechanism 
to  extend  within  the  angle  ABC. 

Motion  of  block  to  be  as  follows: 
J  BA  to  left,  constant  acceleration. 
J  BA  to  left,  constant  velocity. 
I  BA  to  left,  constant  acceleration. 
4  AB  to  right,  increasing    and    decreasing    harmonic    motion. 


KINEMATIC  PROBLEMS 


327 


BA  to  left,  constant  acceleration. 

BA  to  left,  constant  velocity. 

BA  to  left,  constant  acceleration. 

AB  to    right,    increasing   and    decreasing   harmonic    motion. 

BC  up,  constant  acceleration. 

BC  up,  constant  velocity. 

BC  up,  constant  acceleration. 


FIG.  190. 
AB 


i  BC  down,  harmonic  motion. 

J  BC  up,  constant  acceleration. 

J  BC  up,  constant  velocity. 

J  BC  up,  constant  acceleration. 

\  CB  down,  increasing  harmonic  motion. 

f  CB  down,  constant  velocity. 

f  CB  down,  decreasing  harmonic  motion. 

192.  Assignment. — Path  ABC  of  block  as  follower  to  be  a 
groove.     Block  to  move  from  A  to  B  to  C  to  B  to  A. 


B 


FIG.  191. 


No  cams  are  to  be  used  in  the  mechanism,  and  no  part  of  the 
driving  mechanism  is  to  extend  within  the  angle  ABC. 

Rack  and  pinion  or  chain  drive  cannot  be  used  directly  to 
produce  motion. 


328 


MACHINE  DESIGN 


193.  Assignment. — Block  as  follower  to  be  driven  in  groove 
with  center  line  ABC.  No  part  of  mechanism  to  extend  within 
the  angle  ABC.  Use  one  disc  cam  and  one  cylindrical  cam. 
No  bell  cranks  or  pivoted  levers  can  be  employed.  Motion  of 
block  to  be  as  follows: 

^  BA,  constant  acceleration. 
\  BA,  constant  velocity. 
I  BA,  constant    acceleration, 
f  AB,  harmonic   motion  increasing. 
£  AB,  constant  velocity, 
f  AB,  harmonic  motion  decreasing. 
J  BC,  constant  acceleration. 
J  BC,  constant  velocity. 
J  BC,  constant  acceleration. 
CB,  harmonic  motion. 


FIG.  192. 


194.  Assignment.  —  Block  as  follower  to  move  in  groove  whose 
center  line  is  ABC.  Driven  by  a  single  cylindrical  cam,  which 
may,  however,  carry  more  than  one  groove.  Not  more  than 
two  levers  or  bell  cranks  and  two  connecting  rods  may  be  used. 

No  part  of  mechanism  to  extend  within  the  angle  ABC  and 
the  cam  itself  must  be  located  in  the  angle  DBE. 

Motion  of  block  to  be  as  follows: 

J  A  to  B}  constant  acceleration. 
|  A  to  B,  constant  velocity. 
I  A  to  B,  constant  acceleration. 
|  B  to  C,  increasing  harmonic. 
I  B  to  C,  constant  velocity. 
|  B  to  C,  decreasing  harmonic. 
J  C  to  B,  constant  acceleration. 


KINEMATIC  PROBLEMS 

\  C  to  B,  constant  velocity. 
\  C  to  B,  constant  acceleration. 
J  B  to  A,  increasing  harmonic. 
4  B  to  A,  constant  velocity. 
\  B  to  A,  decreasing  harmonic. 


329 


.. E 


\D 
FIG.  193. 


195.  Assignment.  —  Block  as  follower  to  be  driven  in  groove 
with  center  line  ABC.  No  part  of  mechanism  to  extend  within 
the  angle  ABC.  Use  one  disc  cam  and  one  cylindrical  cam. 
No  bell  cranks  or  pivoted  levers  may  be  employed. 


FIG.  194. 


Motion  to  be  as  follows: 
J  BA,  constant  acceleration. 
J  BA}  constant  velocity. 
^  BA,  constant  acceleration  (decreasing). 
|  AB,  harmonic  motion  (increasing). 
J  A  B,  constant  velocity. 
f  A  B,  harmonic  motion  (decreasing). 

BC,  same  motion  as  given  in  the  first  three  above  for  B  to  A. 

CB}  with    harmonic  motion  (increasing  and  decreasing). 


330 


MACHINE  DESIGN 


196.  Assignment. — Design  a  mechanism  to  drive  a  block  over 
the  path  ABCDCBA. 


FIG.  195. 


-D 


ABC  ED  is  center  line  of  groove. 
Use  no  cams,  no  chains,  no  racks. 
Single  rotating  shaft. 

197.  Assignment.—  Path  ABC  to  be  a  groove.     CB  = 

Block  to  be  driven  in  this  groove. 

Motion  to  be  obtained  from  a  single  disc  cam  and  no  part  of 
the  mechanism,  excepting  a  single  guiding  arm  to  impart  motion 
to  block,  to  extend  without  the  angle  ABD  at  any  time  during 
the  period  of  motion.  Use  not  more  than  two  levers  or  bell 
cranks  and  no  connecting  links. 

Blocks  to  make  complete  cycle  in  one  revolution  of  cam. 


B 


D 


FIG.  196. 


Motion  of  block  to  be  as  follows : 


J  B  to  A,  uniform  acceleration, 
f  B  to  A,  uniform  motion. 
J  B  to  A,  uniform  acceleration. 
A  to  B,  harmonic  motion. 


KINEMATIC  PROBLEMS 


331 


}  B  to  C,  uniform  acceleration, 
f  B  to  C,  uniform  motion. 
J  B  to  C,  uniform  acceleration. 
C  to  B,  harmonic  motion. 

198.  Assignment. — Construct  cams  and  mechanism  to  drive  a 
point  over  the  path  ABC  DC  E  and  reverse.  Use  but  one  rotat- 
ing shaft  and  not  more  than  two  cams,  either  disc  or  cylindrical. 
No  part  of  the  mechanism  to  extend  within  the  angle  ABD,  or 
above  the  line  DF. 


D 


A 


FIG.  197. 
'£  =  *  AB. 


CE  =  BD. 

199.  Assignment. — A  BCD  A  is  the  center  line  of  a  groove. 
Design  a  mechanism  to  drive  a  square  block  over  the  groove. 
Use  one  rotating  shaft,  0. 

There  should  be  no  opportunity  for  block  to  wedge  at  corners. 
Use  no  cams,  chains,  racks  or  screws. 


D 


FIG.  198. 


Following  along  the  line  of  the  above  assignment,  others  may 
be  made  referring  to  a  base  runner  around  the  base-ball  diamond 


332 


MACHINE  DESIGN 


using  various  forward  and  backward  movements,  and  various 
constant  velocities  and  accelerations. 

200.  Assignment. — Required  to  design  the  mechanism  and 
cams  to  produce  some  word  at  the  end  of  the  pencil  arm.  The 
location  of  the  parts  should  be  selected  so  as  to  show  the  univer- 


FIG.  199. 

sality  of  the  cam  motion.  All  dimensions  are  to  be  selected  by 
the  student.  Care  should  be  exercised  that  the  cam  curves  do 
not  slope  at  too  great  an  angle. 

201.  Assignment. — Required  to   design  the   mechanism   and 
cams  to  produce  some  word  at  the  end  of  the  pencil  arm.     The 


KINEMATIC  POBLEHS 


333 


mechanism  is  to  have  but  three  moving  parts.  All  dimensions 
are  to  be  selected  by  the  student.  Care  should  be  exercised  that 
the  cam  curves  do  not  slope  at  too  great  an  angle. 


FIG.  200. 


334 


MACHINE  DESIGN 


202.  Assignment. — Required  to  design  the  mechanism  for  a 
writing  cam  as  shown  in  Fig.  201.  The  sliding  block  X  is  moved 
by  screw  connection.  All  sizes  to  be  selected  by  the  student. 
Such  a  cam  may  be  used  for  outlining  any  simple  figure  in  design 
as  well. 


KINEMATIC  PROBLEMS 
Mechanism  of  the  Rites  Inertia  Governor 


335 


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336 


MACHINE  DESIGN 
Mechanism  of  the  Centrifugal  Governor 


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KINEMATIC  PROBLEMS 
Mechanism  of  the  Straight  Line  Governor 


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Valve  Motions) 


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ANALYSIS 


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ipse  and  fill  in  table  of  events. 


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INDEX 


Abbreviations,  1 

Air  hoist,  299 

Alloy  steels,  14 

Alloys,  15 

Arms  of  gears,  196 

Automobile  clutches,  176 

Ball  bearings,  design,  157 

endurance,  157 

journal,  153 

materials,  156 

step,  155 
Beams,  cast  iron,  28 

formulas,  7 

uniform  strength,  8 
Bearings,  adjustment,  129 

ball,  153 

experiments,  138 

friction,  134,  145 

heating,  136 

Hyatt,  160 

journal,  128 

lubrication,  131 

Mossberg,  164 

multiple,  149 

pressure,  134 

roller,  159 

sliding,  120 

step,  144   . 
Belting,  friction,  221^ 

slip,  223 

speed,  227 

strength,  225 

width,  226 
Bevel  shear,  plain,  287  j 

rotary,  296 
Bobbin  winder,  313 
Boiler,  shells,  50 

tubes,  55 
Bolts  and  nuts,  91 


Brass,  16 
Bronze,  16 
Bulldozer,  293 
Butt  joints,  99 

Cams,  accelerating,  324 

conical,  321 

crosshead,  320 

lever,  319 

planer,  311 

sewing  machine,  312 

steamboat,  314 

writing,  332 
Caps  and  bolts,  142 
Castings,  iron,  10 

steel,  14 

Chain  drives,  197 
Clip  former,  324 
Clutches,  173 

automobile,  176 

press,  280 
Columns,  4 
Cotters,  104 
Cotton  ropes,  231 
Coupling  bolts,  178 
Couplings,  171. 
Crane  hooks,  94 
Cranks  and  levers,  200 
Crucible  steel,  13 
Curved  frames,  design,  47 

tests,  42 
Cylinders,  steam,  78 

tests,  80 

thick,  51 

Die  heads,  punch,  283 

sliding,  251 

stationary,  253 
Discs,  conical,  216 

logarithmic,  217 


339 


340 


INDEX 


Discs,  plain,  215 

tests,  218 
Drawings,  size  and  scale,  236 

Elliptic  springs,  114 

Factors  of  safety,  17 
Fittings,  pipe,  flanged,  71 

screwed,  69 
Flanged  fittings,  71 
Flanging  machine,  297 
Flat  plates,  83 

springs,  113 
Fly  wheel,  experiments,  207 

press,  274 

rims,  204 

safe  speeds,  205 
Formulas,  3 
Frames,  curved,  42 

design,  21 

machine,  26 

press,  255 

riveter,  40 

shape,  38 

shear,  45,  266 

stresses,  39 
Friction,  belting,  221 

journals,  139 
,  145 


Gears,  bevel,  195 

cut,  189 

design,  310 

rim  and  arms,  196 

speed,  212 

teeth,  186 
Governor,  centrifugal,  336 

shaft,  337 

Hangers,  182 
Heating  of  journals,  136 
Helical  springs,  107 
Hoist,  air,  299 
Hooks,  crane,  94 
Hoops,  steel,  68 
Hyatt  bearings,  160 


Iron,  cast,  10 

malleable,  11 
wrought,  12 

Joint  pins,  104 
Joints,  rim,  210 
Joints,  riveted,  butt,  99 

diamond,  103 

lap,  98 

tube,  67 
Journals,  128 

strength  of,  142 

Keys,  shafting,  178 
Kinematics  of  machines,  309 

Lap  joints,  98 

Lever  design,  240 

Link  motion,  Stephenson,  338 

Walschaert,  339 
Lubrication,  131 

Machine  frames,  26 

screws,  93 

supports,  25 
Malleable  iron,  11 
Manganese,  bronze,  16 

steel,  14 

Manila  ropes,  229 
Materials,  9 
Metals,  strength  of,  18 
Mossberg  bearings,  165 
Mushet  steel,  15 

Nickel  steel,  14 
Notation,  2 

Open  hearth  steel,  13 

Phosphor  bronze,  16 
Pipe,  fittings,  69 

tables,  56 
Pivots,  conical,  146 

flat,  145  . 
-  Schiele's,  147 
Plates,  flat,  83 

steel,  87 


INDEX 


341 


Power  press,  294 
Press,  foot  power,  262 

hand  power,  261 

power,  294 

shear,  295 

toggle  joint,  235 
Pulleys  for  press,  273 
Punch,  hand  power,  263 

horizontal,  288 

Quick  return,  315 

Riveted  joints,  96 
Riveter,  Allen,  300 

alligator,  304 

frames,  40 

Hanna,  302 

hydraulic,  307 

lever,  306 

mudring,  305 

Riveting  machine,  hydraulic,  308 
Roller  bearings,  159 

design,  163 

step,  162 
Rope  transmission,  cotton,  231 

Manila,  229 

strength,  230 

wire,  232 
Rotary  shear,  295 

Schiele's  pivot,  147 
Screw,  design,  246 

machine,  93 
Sections,  cast  iron,  27 
Shaft  for  press,  276 
Shafting,  bending,  169 

clutches,  173 

couplings,  171 

diameter,  168 

hangers,  182       .  v 

keys,  178 

Shapes  of  frames,  38 
Shear  press,  clutches,  280 

die  head,  283 

fly  wheel,  274 

forces,  272 

frame,  45,  266 


Shear  press,  gears,  277 

pulleys,  273 

shaft,  276 

sliding  head,  279 

types,  284 
Shear,  rotary,  295 
Shells,  thin,  50 

thick,  51 

Silent  chains,  199 
Slides,  angular,  120 

circular,  124 

flat,  122 

gibbed,  122 
Slip  of  belts,  223 
Springs,  elliptic,  114 

experiments,  109 

flat,  113 

helical,  107 

torsion,  112 
Sprocket  wheels,  197 
Standard  for  press,  247 
Steam  cylinders,  78 
Steel,  alloys,  14 

Bessemer,  13 

castings,  14 

crucible,  13 

mushet,  15 

open  hearth,  13 

plates,  87 
Step  bearings,  144 
Stephenson  link  motion,  338 
Strength  of  materials,  17 
Stuffing  boxes,  124 
Supports,  machine,  25 

Tests  of  gears,  experiments,  191 

formulas,  190 

practice,  193 

proportions,  187 

strength,  188 
Thrust  bearings,  150 
Toggle  joint  press,  alternate  design, 
260 

analysis,  239 

design  No.  1,  235 

die  head,  250 

frame,  255 


342  INDEX 

Toggle  lever,  240  Units,  1 

screw,  246 

,     ,   n.«  Vanadium  steel,  15 
standard,  247 

toggle,  248  Walschaert  link  motion,  339 

Torsion  springs,  112  Wire  ropes,  232 

Tubes,  boiler,  55  Wooden  pulleys,  212 

joints,  67  Writing  cam,  332 

tests  on,  62  Wrought  iron,  12 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


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6    193} 


261931 


IMA 
MAY  7  1941  M 

MAY  35  1948 


LD  21-2m-l,'33  (52m) 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


